path: root/Master/texmf-dist/doc/generic/polexpr/polexpr-ref.rst.txt

.. comment: -*- fill-column: 72; mode: rst -*-

===================
 polexpr reference
===================

.. _quick:

Syntax overview via examples
----------------------------

The syntax to define a new polynomial is::

    \poldef polname(x):= expression in variable x;

..

The expression will be parsed by the services of xintexpr_, with some
polynomial aware functions added to its syntax; they are described in
detail :ref:`below <polexpr08>`.  The parser accepts and will handle
exactly arbitrarily big integers or fractions.

.. note::

   xintexpr_ does not automatically reduce fractions to lowest terms,
   and, so far (but this may change in future) neither does :ref:`\\poldef
   <poldef;>`.
   See :ref:`rdcoeffs() <rdcoeffs>` and the macro
   :ref:`\\PolReduceCoeffs <PolReduceCoeffs>`.


- In place of ``x`` an arbitrary *dummy variable* is authorized,
  i.e. per default one ``a, .., z, A, .., Z`` (more letters can be declared
  under Unicode engines).

- ``polname`` is a word (no space) built with *letters*, *digits*, and
  the ``@``, ``_`` and ``'`` characters are allowed.  The polynomial
  name **must** start with a letter.

  For guidelines regarding ``_`` and ``@`` see Technicalities_.

- The colon before the equality sign is optional and its (reasonable)
  catcode does not matter.

- The semi-colon at the end of the expression is mandatory. It is not
  allowed to arise from expansion (despite the fact that the expression
  itself will be parsed using only expansion), it must be "visible"
  immediately.

There are some potential problems (refer to the Technicalities_ section at
bottom of this page) with the semi-colon as expression terminator, so an
alternative syntax is provided, which avoids it altogether::

    \PolDef[optional letter]{<polname>}{<expr. using letter as indeterminate>}

The ``\PolDef`` optional first argument defaults to ``x`` and must be
used as the indeterminate in the expression.

Examples:

``\poldef f(x):= 1 - x + quo(x^5,1 - x + x^2);``

``\PolDef{f}{1 - x + quo(x^5,1 - x + x^2)}``
    Both parse the polynomial
    expression, and they create internally macros serving to
    incarnate the polynomial, its coefficients, and the associated
    polynomial function.

    The polynomial can then be used in further polynomial definitions,
    be served as argument to package macros, or appear as a variable in
    various functions `described later <polexpr08_>`_.

    .. warning::

       Both the function ``quo()`` (as shown in the example above), and
       the infix operator ``/`` are mapped to the Euclidean quotient.

       This usage of ``/`` to stand for the Euclidean quotient is
       **deprecated** and reserved for a (somewhat improbable) possible
       extension of the package to handle rational functions as well.

    .. _warningtacit:

    .. attention::

       Tacit multiplication rules let the parser when encountering
       ``1/2 x^2`` skip the space and thus handle it as ``1/(2*x^2)``.
       But then it gives zero, because `/` stands for the Euclidean
       quotient operation here.

       Thus one must use ``(1/2)x^2`` or ``1/2*x^2`` or
       ``(1/2)*x^2`` for disambiguation: ``x - 1/2*x^2 + 1/3*x^3...``. It is
       simpler to move the denominator to the right: ``x - x^2/2 +
       x^3/3 - ...``.

       It is worth noting that ``1/2(x-1)(x-2)`` suffers the same issue:
       xintexpr_\ 's tacit multiplication always "ties more", hence this
       gets interpreted as ``1/(2*(x-1)*(x-2))`` not as
       ``(1/2)*(x-1)*(x-2)`` and then gives zero by
       polynomial division. Thus, in such cases, use one of
       ``(1/2)(x-1)(x-2)``, ``1/2*(x-1)(x-2)`` or ``(x-1)(x-2)/2``.

    ``\poldef P(x):=...;`` defines ``P`` as a *polynomial function*,
    which can be used inside ``\xinteval``, as::

      \xinteval{P(3 + 7 + 11)}

    or even as::

      \xinteval{P(Q1 + Q2 + Q3)}

    where ``Q1``, ``Q2``, ``Q3`` are polynomials.  The evaluation result,
    if not a scalar, will then be printed as ``pol([c0,c1,...])`` which
    stands for a polynomial variable having the listed coefficients; see
    :ref:`pol() <pol>`.

    Indeed, as seen above with ``Q1``, the symbol ``P`` also stands for
    a *variable of polynomial type*, which serves as argument to
    polynomial specific functions such as :ref:`deg() <deg>` or
    :ref:`polgcd() <polgcd>`, or as argument to other polynomials (as
    above), or even simply stands for its own in algebraic expressions
    such as::

      \poldef Q(z):= P^2 + z^10;

    Notice that in the above, the ``(z)`` part is mandatory, as it informs
    ``\poldef`` of the letter used for the indeterminate.  In the above
    ``P(z)^2`` would give the same as ``P^2`` but the latter is slightly
    more efficient.

    One needs to acquire a good understanding of when the symbol ``P``
    will stand for a function and when it will stand for a variable.

    - If ``P`` and
      ``Q`` are both declared polynomials then::

        (P+Q)(3)%  <--- attention, does (P+Q)*3, not P(3)+Q(3)

      is currently evaluated as ``(P+Q)*3``, because ``P+Q`` is not
      known as a *function*, but *only as a variable of polynomial
      type*.  Note that :ref:`evalp(P+Q,3) <evalp>` gives as expected
      the same as ``P(3)+Q(3)``.

    - Also::

        (P)(3)%  <--- attention, does P*3, not P(3)

      will compute ``P*3``, because one can not in current xintexpr_ syntax
      enclose a function name in parentheses: consequently it is the variable
      which is used here.

    There is a *meager possibility* that in future some internal changes
    to xintexpr_ would let ``(P)(3)`` actually compute ``P(3)`` and
    ``(P+Q)(3)`` compute ``P(3) + Q(3)``, but note that ``(P)(P)`` will
    then do ``P(P)`` and not ``P*P``, the latter, current
    interpretation, looking more intuitive. Anyway, do not rely too
    extensively on tacit ``*`` and use explicit ``(P+Q)*(1+2)`` if this
    is what is intended.


``\PolLet{g}={f}``
    saves a copy of ``f`` under name ``g``. Also usable without ``=``.

    Has exactly the same effect as ``\poldef g(x):=f;`` or ``\poldef
    g(w):=f(w);``\ .

``\poldef f(z):= f^2;``
    redefines ``f`` in terms of itself.  Prior to ``0.8`` one needed
    the right hand side to be ``f(z)^2``.  Also, now ``sqr(f)``  is
    possible (also ``sqr(f(x))`` but not ``sqr(f)(x)``).

    It may look strange that an indeterminate variable is used on
    left-hand-side even though it may be absent of right-hand-side, as
    it seems to define ``f`` always as a polynomial function.

    This is a legacy of pre-``0.8`` context.

    .. important::

       Note that ``f^2(z)`` or ``sqr(f)(z)`` will give a logical but
       perhaps unexpected result: first ``f^2`` is computed, then the
       opening parenthesis is seen which inserts a tacit multiplication
       ``*``, so in the end it is as if the input had been ``f^2 * z``.
       Although ``f`` is both a variable and a function, ``f^2`` is
       computed as a polynomial *variable* and ceases being a function.

``\poldef f(T):= f(f);``
    again modifies ``f``.  Here it is used both as variable and as
    a function.  Prior to ``0.8`` it needed to be ``f(f(T))``.

``\poldef k(z):= f-g(g^2)^2;``
    if everybody followed, this should now define the zero polynomial...
    And ``f-sqr(g(sqr(g)))`` computes the same thing.

    We can check this in a typeset document like this::

        \poldef f(x):= 1 - x + quo(x^5,1 - x + x^2);%
        \PolLet{g}={f}%
        \poldef f(z):= f^2;%
        \poldef f(T):= f(f);%
        \poldef k(w):= f-sqr(g(sqr(g)));%
        $$f(x) = \vcenter{\hsize10cm \PolTypeset{f}} $$
        $$g(z) = \PolTypeset{g} $$
        $$k(z) = \PolTypeset{k} $$
        \immediate\write128{f(x)=\PolToExpr{f}}% ah, here we see it also

``\poldef f'(x):= diff1(f);``
    (new at ``0.8``)

``\PolDiff{f}{f'}``
    Both set ``f'`` (or any other chosen name) to the derivative
    of ``f``.

    .. important::

       This is not done automatically. If some new definition needs to use
       the derivative of some available polynomial, that derivative
       polynomial must have been previously defined: something such as
       ``f'(3)^2`` will not work without a prior definition of ``f'``.

       But one can now use ``diff1(f)`` for on-the-spot construction with no
       permanent declaration, so here ``evalp(diff1(f),3)^2``. And
       ``diff1(f)^2`` is same as ``f'^2``, assuming here ``f'`` was declared
       to be the derived polynomial.

       Notice that the name ``diff1()`` is experimental and may change. Use
       ``\PolDiff{f}{f'}`` as the stable interface.

``\PolTypeset{P}``
    Typesets (switching to math mode if in text mode)::

       \poldef f(x):=(3+x)^5;%
       \PolDiff{f}{f'}\PolDiff{f'}{f''}\PolDiff{f''}{f'''}%
       $$f(z)   = \PolTypeset[z]{f}    $$
       $$f'(z)  = \PolTypeset[z]{f'}   $$
       $$f''(z) = \PolTypeset[z]{f''}  $$
       $$f'''(z)= \PolTypeset[z]{f'''} $$

    See `its documentation <PolTypeset_>`_ for the configurability
    via macros.

    Since ``0.8`` `\\PolTypeset <PolTypeset_>`_ accepts directly an
    expression, it does not have to be a pre-declared polynomial name::

        \PolTypeset{mul(x-i,i=1..5)}

``\PolToExpr{P}``
    Expandably (contrarily to `\\PolTypeset <PolTypeset_>`_)
    produces ``c_n*x^n + ... + c_0`` starting from the leading
    coefficient.  The ``+`` signs are omitted if followed by negative
    coefficients.

    This is useful for console or file output.  This syntax is Maple and
    PSTricks ``\psplot[algebraic]`` compatible; and also it is
    compatible with ``\poldef`` input syntax, of course.  See
    `\\PolToExprCaret`_ for configuration of the ``^``, for example to
    use rather ``**`` for Python syntax compliance.

    Changed at ``0.8``: the ``^`` in output is by default of catcode 12
    so in a draft document one can use ``\PolToExpr{P}`` inside the
    typesetting flow (without requiring math mode, where the ``*`` would
    be funny and ``^12`` would only put the ``1`` as exponent anyhow;
    but arguably in text mode the ``+`` and ``-`` are not satisfactory
    for math, except sometimes in monospace typeface, and anyhow TeX is
    unable to break the expression across lines, barring special help).

    See :ref:`\\PolToExpr{\<pol. expr.\>} <PolToExpr>` and related macros for customization.

    Extended at ``0.8`` to accept as argument not only the name of a
    polynomial variable but more generally any polynomial expression.


Using defined polynomials in floating point context
---------------------------------------------------

Exact manipulations with fractional coefficients may quickly lead to
very large denominators.  For numerical evaluations, it is advisable
to a use a floating point context.  But for the polynomial to be
usable as a function in floating point context, an extra step beyond
``\poldef`` is required: see `\\PolGenFloatVariant`_.  Then the
``\xintfloateval`` macro from xintexpr_ will recognize the polynomial
as a genuine function (with already float-rounded coefficients, and
using a Horner scheme).

But `\\PolGenFloatVariant`_ must be used each time the polynomial gets
redefined or a new polynomial is created out of it.  Functions such as
for example :ref:`deg() <deg>` which handle the polynomial as an entity
are only available within the ``\poldef`` and ``\xinteval`` (or
``\xintexpr``) parsers.  Inside ``\xintfloateval`` a polynomial can only
serve as a numerical function (and only after declaration via
`\\PolGenFloatVariant`_), and not as a variable.

In some cases one may wish to replace a polynomial having acquired
very big fractional coefficients with a new one whose coefficients
have been float-rounded.  See :ref:`\\PolMapCoeffs <PolMapCoeffs>`
which can be used for example with the ``\xintFloat`` macro from the
xintfrac_ package to achieve this.


.. _polexpr08:

The polexpr ``0.8`` extensions to the ``\xintexpr`` syntax
----------------------------------------------------------

All the syntax elements described in this section can be used in the
``\xintexpr/\xinteval`` context (where polynomials can be obtained from
the ``pol([])`` constructor, once polexpr is loaded):  their usage is
not limited to only ``\poldef`` context.

.. note::

   If a variable ``myPol`` defined via ``\xintdefvar`` turns out
   to be a polynomial, the difference with those declared via ``\poldef``
   will be:

   1. ``myPol`` is not usable as *function*, but only as a variable.
      Attention that ``f(x)`` if ``f`` is only a variable (even a
      polynomial one) will actually compute ``f * x``.

   2. ``myPol`` is not known to the polexpr package, hence for example the
      macros to achieve localization of its roots are unavailable.

      In a parallel universe I perhaps have implemented this expandably
      which means it could then be accessible with syntax such as
      ``rightmostroot(pol([42,1,34,2,-8,1]))`` but...


Warning about unstability of the new syntax
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. warning::

   Consider the entirety of this section as **UNSTABLE** and
   **EXPERIMENTAL** (except perhaps regarding ``+``, ``-`` and ``*``).

   And this applies even to items not explicitly flagged with one of
   **unstable**, **Unstable**, or **UNSTABLE** which only reflect that
   documentation was written over a period of time exceeding one minute,
   enough for the author mood changes to kick in.

   It is hard to find good names at the start of a life-long extension
   program of functionalities, and perhaps in future it will be
   preferred to rename everything or give to some functions other
   meanings.  Such quasi-complete renamings happened already a few times
   during the week devoted to development.


Infix operators ``+, -, *, /, **, ^``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   As has been explained in the `Syntax overview via examples`_
   section these infix operators have been made polynomial aware, not
   only in the ``\poldef`` context, but generally in any
   ``\xintexpr/\xinteval`` context, inclusive of ``\xintdeffunc``.

   Conversely functions declared via ``\xintdeffunc`` and making use of
   these operators will automatically be able to accept polynomials
   declared from ``\poldef`` as variables.

   Usage of ``/`` for euclidean division of polynomials is **deprecated**.
   Only in case of a scalar denominator is it to be considered stable.
   Please use rather ``quo()``.

Experimental infix operators ``//, /:``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   Here is the tentative behaviour of ``A//B`` according to types:

   - ``A`` non scalar and ``B`` non scalar: euclidean quotient,

   - ``A`` scalar and ``B`` scalar: floored division,

   - ``A`` scalar and ``B`` non scalar: produces zero,

   - ``A`` non scalar and ``B`` scalar: coefficient per
     coefficient floored division.

   This is an **experimental** overloading of the ``//`` and ``/:``
   from ``\xintexpr``.

   The behaviour in the last case, but not only, is to be considerd
   **unstable**. The alternative would be for ``A//B`` with ``B``
   scalar to act as ``quo(A,B)``.  But, we have currently chosen to let
   ``//B`` for a scalar ``B`` act coefficient-wise on the numerator.
   Beware that it thus means it can be employed with the idea of doing
   euclidean division only by checking that ``B`` is non-scalar.

   The ``/:`` operator provides the associated remainder so always
   ``A`` is reconstructed from ``(A//B)*B + A/:B``.

   If ``:`` is active character use ``/\string:`` (it is safer to use
   ``/\string :`` if it is not known if ``:`` has catcode other, letter,
   or is active, but note that ``/:`` is fine and needs no precaution if
   ``:`` has catcode letter, it is only an active ``:`` which is
   problematic, like for all other characters possibly used in an
   expression).

     **UNSTABLE**

     As explained above, there are (among other things) hesitations
     about behaviour with ``pol2`` a scalar.

Comparison operators ``<, >, <=, >=, ==, !=``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   **NOT YET IMPLEMENTED**

   As the internal representation by xintfrac_ and xintexpr_ of
   fractions does not currently require them to be in reduced terms,
   such operations would be a bit costly as they could not benefit from
   the ``\pdfstrcmp`` engine primitive.  In fact xintexpr_ does not use
   it yet anywhere, even for normalized pure integers, although it could
   speed up signifcantly certain aspects of core arithmetic.

   Equality of polynomials can currently be tested by computing the
   difference, which is a bit costly.  And of course the ``deg()``
   function allows comparing degrees.  In this context note the
   following syntax::

     (deg(Q)) ?? { zero } { non-zero scalar } { non-scalar }

   for branching.


.. _pol:

``pol(<nutple expression>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   This converts a nutple ``[c0,c1,...,cN]`` into the polynomial
   variable having these coefficients.  Attention that the square
   brackets are **mandatory**, except of course if the argument is
   actually an expression producing such a "nutple".

      Currently, this process will not normalize the coefficients (such
      as reducing to lowest terms), it only trims out the leading zero
      coefficients.

   Inside ``\xintexpr``, this is the only (allowed) way to create ex
   nihilo a polynomial variable; inside ``\poldef`` it is an alternative
   input syntax which is more efficient than the input ``c0 + c1 * x + c2 *
   x^2 + ...``.

.. important::

   Whenever an expression with polynomials collapses to a constant, it
   becomes a scalar.  There is currently no distinction during the
   parsing of expressions by ``\poldef``
   or ``\xintexpr`` between constant polynomial variables and scalar
   variables.

   Naturally, ``\poldef`` can be used to declare a constant polynomial
   ``P``, then ``P`` can also be used as function having a value
   independent of argument, but as a variable, it is non-distinguishable
   from a scalar (of course functions such as ``deg()`` tacitly
   consider scalars to be constant polynomials).
   
   Notice that we tend to use the vocable "variable" to refer to
   arbitrary expressions used as function arguments, without implying
   that we are actually referring to pre-declared variables in the sense
   of ``\xintdefvar``.

.. _lpol:

``lpol(<nutple expression>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   This converts a nutple ``[cN,...,c1,c0]`` into the polynomial
   variable having these coefficients, with leading coefficients coming
   first in the input.  Attention that the square brackets are
   **mandatory**, except of course if the argument is actually an
   expression producing such a "nutple".

      Currently, this process will not normalize the coefficients (such
      as reducing to lowest terms), it only trims out the leading zero
      coefficients.

      **NAME UNSTABLE**

   It can be used in ``\poldef`` as an alternative input syntax, which
   is more efficient than using the algebraic notation with monomials.

   (new with ``0.8.1``, an empty nutple will cause breakage)

.. _xintevalpolexpr:

``\xinteval{<pol. expr.>}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~

   This is documented here for lack of a better place: it evaluates the
   polynomial expression then outputs the "string" ``pol([c0, c1, ..., cN])``
   if the degree ``N`` is at least one (and the usual scalar output else).

   The "pol" word uses letter catcodes, which is actually mandatory for
   this output to be usable as input, but it does not make sense to use
   this inside ``\poldef`` or ``\xintexpr`` at it means basically
   executing ``pol(coeffs(..expression..))`` which is but a convoluted
   way to obtain the same result as ``(..expression..)`` (the
   parentheses delimiting the polynomial expression).

   For example, ``\xinteval{(1+pol([0,1]))^10}`` expands (in two steps)
   to::

     pol([1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1])

   You do need loading polexpr for this, else of course ``pol([])``
   remains unknown to ``\xinteval{}`` as well as the polynomial algebra !
   This example can also be done as
   ``\xinteval{subs((1+x)^10,x=pol([0,1]))}``.

   I hesitated using as output the polynomial notation as produced by
   `\\PolToExpr{} <poltoexpr_>`_, but finally opted for this.

.. _evalp:

``evalp(<pol. expr.>, <pol. expr>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   Evaluates the first argument as a polynomial function of the
   second.  Usually the second argument will be scalar, but this is not
   required::

     \poldef K(x):= evalp(-3x^3-5x+1,-27x^4+5x-2);

   If the first argument is an already declared polynomial ``P``, use
   rather the functional form ``P()`` (which can accept a numerical as
   well as polynomial argument) as it is more efficient.

   One can also use ``subs()`` syntax [#]_ (see xintexpr_ documentation)::

     \poldef K(x):= subs(-3y^3-5y+1, y = -27x^4+5x-2);

   but the ``evalp()`` will use a Horner evaluation scheme which is
   usually more efficient.

   .. [#] by the way Maple uses the opposite, hence wrong, order
          ``subs(x=..., P)`` but was written before computer science
          reached the xintexpr_ heights.  However it makes validating
          Maple results by polexpr sometimes cumbersome, but perhaps
          they will update it at some point.

   ..

     **name unstable**

     ``poleval``? ``evalpol``?  ``peval``? ``evalp``? ``value``?
     ``eval``? ``evalat``? ``eval1at2``? ``evalat2nd``?
     
     Life is so complicated when one asks questions. Not everybody does,
     though, as is amply demonstrated these days.

     **syntax unstable**

     I am hesitating about permuting the order of the arguments.

.. _deg:

``deg(<pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~~

   Computes the degree.

.. important::

   As ``\xintexpr`` does not yet support infinities, the degree of
   the zero polynomial is ``-1``.  Beware that this breaks additivity
   of degrees, but ``deg(P)<0`` correctly detects the zero polynomial,
   and ``deg(P)<=0`` detects scalars.

``coeffs(<pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~~~~~

   Produces the nutple ``[c0,c1,...,cN]`` of coefficients. The highest
   degree coefficient is always non zero (except for the zero
   polynomial...).

     **name unstable**

     I am considering in particular using ``polcoeffs()`` to avoid
     having to overload ``coeffs()`` in future when matrix type
     will be added to xintexpr_.

.. _lcoeffs:

``lcoeffs(<pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~~~~~~

   Produces the nutple ``[cN,....,c1,c0]`` of coefficients, starting
   with the highest degree coefficient.

   (new with ``0.8.1``)

``coeff(<pol. expr.>, <num. expr.>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   As expected. Produces zero if the numerical index is negative or
   higher than the degree.

     **name, syntax and output unstable**

     I am hesitating with ``coeff(n,pol)`` syntax and also perhaps using
     ``polcoeff()`` in order to avoid having to overload ``coeff()``
     when matrix type will be added to xintexpr_.

     The current behaviour is at odds with legacy
     :ref:`\\PolNthCoeff{\<polname\>}{\<index\>} <PolNthCoeff>` regarding negative indices.
     Accessing leading or sub-leading coefficients can be done with
     other syntax, see `lc(<pol. expr.>)`_, and in some contexts it
     is useful to be able to rely on the fact that coefficients with
     negative indices do vanish, so I am for time being maintaining this.

.. _lc:

``lc(<pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~

   The leading coefficient. The same result can be obtained from
   ``coeffs(pol)[-1]``, which shows also how to generalize to access
   sub-leading coefficients.  See the xintexpr_ documentation for
   Python-like indexing syntax.

``monicpart(<pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~

   Divides by the leading coefficient, except that ``monicpart(0)==0``.

     **unstable**

     Currently the coefficients are reduced to lowest terms (contrarily
     to legacy behaviour of `\\PolMakeMonic <polmakemonic_>`_), and
     additionally the xintfrac_ ``\xintREZ`` macro is applied which
     extracts powers of ten from numerator or denominator and stores
     them internally separately. This is generally beneficial to
     efficiency of multiplication.

.. _cont:

``cont(<pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~~~

   The (fractional) greatest common divisor of the polynomial
   coefficients. It is always produced as an irreducible (non-negative)
   fraction.  According to Gauss theorem the content of a product is the
   product of the contents.

   .. commentaire 8 avril 2021

      surprenamment après avoir utilisé  `\\PolIContent <PolIContent_>`_
      une fois on peut utiliser `\\PolIContent`_ directement.

      avec docutils 0.16

   ..

     **name and syntax unstable**

     At ``0.8`` it was created as ``icontent()`` to match the legacy
     macro `\\PolIContent <PolIContent_>`_, whose name in 2018 was
     chosen in relation to Maple's function ``icontent()``, possibly
     because at that time I had not seen that Maple also had a
     ``content()`` function.  Name changed at ``0.8.1``.

     It will change syntax if in future multivariate polynomials are
     supported, and ``icontent()`` will then make a come-back.

``primpart(<pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~

   The quotient (except for the zero polynomial) by
   ``cont(<pol. expr.>)``.  This is thus a polynomial with
   integer coefficients having ``1`` as greatest common divisor.  The
   sign of the leading coefficient is the same as in the original.

   And ``primpart(0)==0``.

   The trailing zeros of the integer coefficients are extracted
   into a power of ten exponent part, in the internal representation.
   
``quorem(<pol. expr.>, <pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   Produces a nutple ``[Q,R]`` with ``Q`` the euclidean quotient and
   ``R`` the remainder.

     **name unstable**

     ``poldiv()``?

``quo(<pol. expr.>, <pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   The euclidean quotient.

   The deprecated ``pol1/pol2`` syntax computes the same polynomial.

``rem(<pol. expr.>, <pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   The euclidean remainder. If ``pol2`` is a (non-zero) scalar, this is
   zero.

   There is no infix operator associated to this, for lack of evident
   notation.  Please advise.

   ``/:`` can be used if one is certain that ``pol2`` is of
   degree at least one.  But read the warning about it being unstable
   even in that case.

.. not implemented

  ``spquorem(pol1,pol2)``
  ~~~~~~~~~~~~~~~~~~~~~~~

   Produces a tuple ``[Q,R]`` with the pseudo-quotient and
   pseudo-remainder.  See `prem(pol1, pol2) <prem_>`_ for
   their definitions.

     **NOT IMPLEMENTED**

     I am hesitating returning rather the nutple ``[b^f, Q, R]`` or
     ``[f, Q, R]``.  Note that the number of non-zero coefficients of
     a polynomial ``P`` can be computed as ``add(?(c),c=coeffs(P))``,
     and in this context I am hesitating abstracting a function to
     provide this [#]_.  The usual problem is that I don't know how to
     name the function.

     I am also hesitating providing rather a function returning only
     ``f`` and ``R``, not ``Q``, which for modular computations we don't
     need to carry along.

     .. [#] one can embed ``\xintiiexpr add(?(c),c=coeffs(P))\relax``
            inside ``\xintexpr`` and it will be more efficient for long
            polynomials, but naturally a core implementation using a
            single ``\numexpr`` would be quite more efficient still.

.. _prem:

``prem(<pol. expr. 1>, <pol. expr. 2>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   Produces a nutple ``[m, spR]`` where ``spR`` is the (special) pseudo
   Euclidean remainder.  Its description is:

   - the standard euclidean remainder ``R`` is ``spR/m``

   - ``m = b^f`` with ``b`` equal to the **absolute value** of the
     leading coefficient of ``pol2``,

   - ``f`` is the number of non-zero coefficients in the euclidean
     quotient, if ``deg(pol2)>0`` (even if the remainder vanishes).

     If ``pol2`` is a scalar however, the function outputs ``[1,0]``.

   With these definitions one can show that if both ``pol1`` and
   ``pol2`` have integer coefficients, then this is also the case of
   ``spR``, which makes its interest (and also ``m*Q`` has integer
   coefficients, with ``Q`` the euclidean quotient, if ``deg(pol2)>0``).
   Also, ``prem()`` is computed faster than ``rem()`` for such integer
   coefficients polynomials.

   .. hint::
      
      If you want the euclidean quotient ``R`` evaluated via ``spR/m``
      (which may be faster, even with non integer coefficients) use
      ``subs(last(x)/first(x),x=prem(P,Q))`` syntax as it avoids
      computing ``prem(P,Q)`` twice.  This does the trick both in
      ``\poldef`` or in ``\xintdefvar``.

      However, as is explained in the xintexpr_ documentation, using
      such syntax in an ``\xintdeffunc`` is (a.t.t.o.w) illusory, due to
      technicalities of how ``subs()`` gets converted into nested
      expandable macros.  One needs an auxiliary function like this::

        \xintdeffunc lastoverfirst(x):=last(x)/first(x);
        \xintdeffunc myR(x)=lastoverfirst(prem(x));

      Then, ``myR(pol1,pol2)`` will evaluate ``prem(pol1,pol2)`` only
      once and compute a polynomial identical to the euclidean
      remainder (internal representations of coefficients may differ).
   
   In this case of integer coefficients polynomials, the polexpr
   internal representation of the integer coefficients in the pseudo
   remainder will be with unit denominators only if that was already the
   case for those of ``pol1`` and ``pol2`` (no automatic reduction to
   lowest terms is made prior or after computation).

   Pay attention here that ``b`` is the **absolute value** of the
   leading coefficient of ``pol2``.  Thus the coefficients of the
   pseudo-remainder have the same signs as those of the standard
   remainder.  This diverges from Maple's function with the same name.


``divmod(<pol. expr. 1>, <pol. expr. 2>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   Overloads the scalar ``divmod()`` and associates it  with the
   experimental ``//`` and ``/:`` as extended to the polynomial type.

   In particular when both ``pol1`` and ``pol2`` are scalars, this is
   the usual ``divmod()`` (as in Python) and for ``pol1`` and ``pol2``
   non constant polynomials, this is the same as ``quorem()``.

     **Highly unstable** overloading of ``\xinteval``\ 's  ``divmod()``.

``mod(<pol. expr. 1>, <pol. expr. 2>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   The ``R`` of the ``divmod()`` output. Same as ``R`` of ``quorem()``
   when the second argument ``pol2`` is of degree at least one.

     **Highly unstable** overloading of ``\xinteval``\ 's  ``mod()``.

``polgcd(<pol. expr. 1>, <pol. expr. 2>, ...)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   Evaluates to the greatest common polynomial divisor of all the
   polynomial inputs.  The output is a **primitive** (in particular,
   with integer coefficients) polynomial.  It is zero if and only if all
   inputs vanish.

   Attention, there must be either at least two polynomial variables, or
   alternatively, only one argument which then must be a bracketed list
   or some expression or variable evaluating to such a "nutple" whose
   items are polynomials (see the documentation of the scalar ``gcd()``
   in xintexpr_).

     The two variable case could (and was, during development) have been
     defined at user level like this::

       \xintdeffunc polgcd_(P,Q):=
         (deg(Q))??{P}{1}{polgcd_(Q,primpart(last(prem(P,Q))))};
       \xintdeffunc polgcd(P,Q):=polgcd_(primpart(P),primpart(Q));%

     This is basically what is done internally for two polynomials, up
     to some internal optimizations.

   **UNSTABLE**

   I hesitate between returning a *primitive* or a *monic* polynomial.
   Maple returns a primitive polynomial if all inputs [#]_ have integer
   coefficients, else it returns a monic polynomial, but this is
   complicated technically for us to add such a check and would add
   serious overhead.

   Internally, computations are done using primitive
   integer-coefficients polynomials (as can be seen in the function
   template above).  So I decided finally to output a primitive
   polynomial, as one can always apply ``monicpart()`` to it.

   Attention that this is at odds with behaviour of the legacy
   `\\PolGCD <PolGCD_>`_ (non expandable) macro.

   .. [#] actually, only two polynomial arguments are allowed by Maple's
          ``gcd()`` as far as I know.

``resultant(<pol. expr. 1>, <pol. expr. 2>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   The resultant.

     **NOT YET IMPLEMENTED**

``disc(<pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~~~

   The discriminant.

     **NOT YET IMPLEMENTED**

``polpowmod(<pol. expr. 1>, <num. expr.>, <pol. expr. 2>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   Modular exponentiation: ``mod(pol1^N, pol2)`` in a more efficient
   manner than first computing ``pol1^N`` then reducing modulo ``pol2``.

   Attention that this is using the ``mod()`` operation, whose current
   experimental status is as follows:

   - if ``deg(pol2)>0``, the euclidean remainder operation,
   - if ``pol2`` is a scalar, coefficient-wise reduction modulo ``pol2``.


   **UNSTABLE**

     This is currently implemented at high level via ``\xintdeffunc`` and
     recursive definitions, which were copied over from a scalar example
     in the xintexpr_ manual::

       \xintdeffunc polpowmod_(P, m, Q) :=
              isone(m)?
                  % m=1: return P modulo Q
                  {   mod(P,Q)  }
                  % m > 1: test if odd or even and do recursive call
                  {   odd(m)? {  mod(P*sqr(polpowmod_(P, m//2, Q)), Q) }
                              {  mod(  sqr(polpowmod_(P, m//2, Q)), Q) }
                   }
                ;%
       \xintdeffunc polpowmod(P, m, Q) := (m)?{polpowmod_(P, m, Q)}{1};%

     Negative exponents are not currently implemented.

     For example::

       \xinteval{subs(polpowmod(1+x,100,x^7),x=pol([0,1]))}
       \xinteval{subs(polpowmod(1+x,20,10), x=pol([0,1]))}

     produce respectively::

       pol([1, 100, 4950, 161700, 3921225, 75287520, 1192052400])
       pol([1, 0, 0, 0, 5, 4, 0, 0, 0, 0, 6, 0, 0, 0, 0, 4, 5, 0, 0, 0, 1])

.. perte de temps terrible pourquoi j'écris cela
     When ``pol2`` is as scalar then the degrees of the modular powers
     ``mod(pol1^N, pol2)`` will in general increase linearly in ``N``
     hence become big.  But one can play with modifying the above
     template and nesting two ``mod()``, one with an integer modulus,
     say ``7``, and the other the a monic integer coefficients
     polynomial such as ``Q = x^2+1``.  Then an integer coefficients
     polynomial ``P`` will have an integer coefficient remainder modulo
     ``Q``, and 

.. _rdcoeffs:

``rdcoeffs(<pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~

   This operates on the internal representation of the coefficients,
   reducing them to lowest terms.

     **name HIGHLY undecided**

``rdzcoeffs(<pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~

   This operates on the internal representation of the coefficients,
   reducing them to lowest terms then extracting from numerator
   or denominator the maximal power of ten to store as a decimal
   exponent.

   This is sometimes favourable to more efficient polynomial algebra
   computations.

     **name HIGHLY undecided**

``diff1(<pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~~~~

   The first derivative.

     **name UNSTABLE**

     This name may be used in future to be the partial derivative with
     respect to a first variable.

``diff2(<pol. expr.>)``
~~~~~~~~~~~~~~~~~~~~~~~

   The second derivative.

     **name UNSTABLE**

     This name may be used in future to be the partial derivative with
     respect to a second variable.


``diffn(<pol. expr. P>, <num. expr. n>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   The ``n``\ th derivative of ``P``. For ``n<0`` computes iterated primitives
   vanishing at the origin.

   The coefficients are not reduced to lowest terms.

     **name and syntax UNSTABLE**

     I am also considering reversing the order of the arguments.

``antider(<pol. expr. P>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~

   The primitive of ``P`` with no constant term. Same as ``diffn(P,-1)``.

``intfrom(<pol. expr. P>, <pol. expr. c>)``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   The primitive of ``P`` vanishing at ``c``, i.e. ``\int_c^x P(t)dt``.

   Also ``c`` can be a polynomial... so if ``c`` is monomial ``x``
   this will give zero!

     **UNSTABLE**

     Allowing general polynomial variable for ``c`` adds a bit of
     overhead to the case of a pure scalar.  So I am hesitating
     maintaining this feature whose interest appears dubious.

   .. attention::

      As the two arguments are both allowed to be polynomials, if by
      inadvertance one exchanges the two, there is no error but the
      meaning of ``intfrom(c,P)`` is completely otherwise, as it
      produces ``c*(x - P)`` if ``c`` is a scalar::

        >>> &pol
        pol mode (i.e. function definitions use \poldef)
        >>> P(x):=1+x^2;
        P = x^2+1
            --> &GenFloat(P) lets P become usable as function in fp mode
            --> &ROOTS(P) (resp. &ROOTS(P,N)) finds all rational roots exactly and
                all irrational roots with at least 10 (resp. N) fractional digits
        >>> intfrom(P,1);
        @_1     pol([-4/3, 1, 0, 1/3])
        >>> intfrom(1,P);
        @_2     pol([-1, 1, -1])
        >>> &bye



.. grosse hésitation ici

   ``intto(<pol. expr. P>, <pol. expr. c>)``
   -----------------------------------------

     ``\int_x^c P(t)dt``.

   c'est l'opposé du précédent

   mais le nom pourrait faire penser à \int_0^x plutôt

``integral(<pol. expr. P>, [<pol. expr. a>, <pol. expr. b>])``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   ``\int_a^b P(t)dt``.

   .. warning::

      The brackets here are not denoting an optional argument but a
      *mandatory* nutple argument ``[a, b]`` with *two items*.  No real
      recoverable-from error check is done on the input syntax.  The
      input can be an xintexpr_ variable which happens to be a nutple
      with two items, or any expression which evaluates to such a
      nutple.

   ``a`` and ``b`` are not restricted to be scalars, they are allowed to
   be themselves polynomial variables or even polynomial expressions.

     To compute ``\int_{x-1}^x P(t)dt`` it is more efficient to use
     ``intfrom(x-1)``.

     Similary to compute ``\int_x^{x+1} P(t)dt``, use ``-intfrom(x+1)``.

     **UNSTABLE**

     Am I right to allow general polynomials ``a`` and ``b`` hence add
     overhead to the pure scalar case ?



Non-expandable macros
---------------------

.. note::

   At ``0.8`` ``polexpr`` is usable with Plain TeX and not only with
   LaTeX.  Some examples given in this section may be using LaTeX syntax
   such as ``\renewcommand``.

.. _poldef;:

``\poldef polname(letter):= expression using the letter as indeterminate;``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This evaluates the *polynomial expression* and stores the
    coefficients in a private structure accessible later via other
    package macros, used with argument ``polname``.  Of course the
    *expression* can make use of previously defined polynomials.

    Polynomial names must start with a letter and are constituted of
    letters, digits, underscores, the ``@`` (see Technicalities_) and
    the right tick ``'``.

    The whole xintexpr_ syntax is authorized, as long as the final
    result is of polynomial type::

       \poldef polname(z) := add((-1)^i z^(2i+1)/(2i+1)!, i = 0..10);

    With fractional coefficients, beware the `tacit multiplication issue
    <warningtacit_>`_.

    Furthermore:

    - a variable ``polname`` is defined which can be used in ``\poldef``
      as well as in ``\xinteval`` for algebraic computations or as
      argument to polynomial aware functions,

    - a function ``polname()`` is defined which can be used in ``\poldef``
      as well as in ``\xinteval``. It accepts there as argument scalars
      and also other polynomials (via their names, thanks to previous
      item).

    Any function defined via ``\xintdeffunc`` and only algebraic
    operations, as well as ople indexing or slicing operations, should
    work fine in ``\xintexpr/\xinteval`` with such polynomial names as
    argument.

    In the case of a constant polynomial, the xintexpr_ *variable* (not
    the internal data structure on which the package macros operate)
    associated to it is indistinguishable from a scalar, it is actually
    a scalar and has lost all traces from its origins as a polynomial
    (so for example can be used as argument to the ``cos()`` function).
    The *function* on the other hand remains a one-argument function,
    which simply has a constant value.

    .. attention::

       The function ``polname()`` is defined **only** for
       ``\xintexpr/\xinteval``
       context.  It will be unknown to ``\xintfloateval``.

       Worse, a
       previously existing floating point function of the same name will
       be made undefined again, to avoid hard to debug mismatches between
       exact and floating point polynomials. This also applies when the
       polynomial is produced not via ``\poldef`` or ``\PolDef`` but
       as result of usage of the other package macros.

       See :ref:`\\PolGenFloatVariant{\<polname\>} <PolGenFloatVariant>` to generate a **function**
       usable in ``\xintfloateval``.

    .. attention::

       Using the **variable** ``mypol`` inside ``\xintfloateval`` will
       generate low-level errors because the infix operators there are
       not polynomial-aware, and the polynomial specific functions such
       as ``deg()`` are only defined for usage inside ``\xintexpr``.

       In short, currently polynomials defined via ``polexpr`` can
       be used in floating point context only for numerical evaluations,
       via **functions** obtained from :ref:`\\PolGenFloatVariant{\<polname\>} <PolGenFloatVariant>`
       usage.

       Changes to the original polynomial via package macros are not
       automatically mapped to the numerical floating point evaluator
       which must be manually updated as necessary when the original
       rational coefficient polynomial is modified.

    The original expression is lost after parsing, and in particular the
    package provides no way to typeset it (of course the package
    provides macros to typeset the computed polynomial).  Typesetting
    the original expression has to be done manually, if needed.

.. _PolDef:

``\PolDef[<letter>]{<polname>}{<expr. using the letter as indeterminate>}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Does the same as `\\poldef <poldef;_>`_ in an undelimited macro
    format, the main interest is to avoid potential problems with the
    catcode of the semi-colon in presence of some packages. In absence
    of a ``[<letter>]`` optional argument, the variable is assumed to be
    ``x``.


.. _PolGenFloatVariant:

``\PolGenFloatVariant{}``
~~~~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolGenFloatVariant{<polname>}``

    Makes the polynomial also usable in the
    ``\xintfloatexpr/\xintfloateval`` parser.  It will therein evaluates
    via an Horner scheme using polynomial coefficients already
    pre-rounded to the float precision.

    See also :ref:`\\PolToFloatExpr{\<pol. expr.\>} <PolToFloatExpr>`.

    .. attention::

       Any operation, for example generating the derivative polynomial,
       or dividing two polynomials or using the ``\PolLet``, must be
       followed by explicit usage of ``\PolGenFloatVariant{<polname>}`` if
       the new polynomial is to be used in ``\xintfloateval``.

.. _PolTypeset:

``\PolTypeset[]{}``
~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolTypeset[<letter>]{<pol. expr.>}``

    Typesets in descending powers, switching to math mode if in text
    mode, after evaluating the polynomial expression::

      \PolTypeset{mul(x-i,i=1..5)}% possible since polexpr 0.8

    The letter used in the input is by default assumed to be ``x``,
    but can be modified by a redefinition of `\\PolToExprInVar`_.

    The letter used in the output is also by default ``x``.  This one
    can be changed on-the-fly via the optional ``<letter>``::

      \PolTypeset[z]{polname or polynomial expression}

    By default zero coefficients are skipped (use ``\poltypesetalltrue``
    to get all of them in output).

    The following macros (whose meanings will be found in the package code)
    can be re-defined for customization. Their default definitions are
    expandable, but this is not a requirement.

.. _PolTypesetCmd:

``\PolTypesetCmd{}``
^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolTypesetCmd{<raw_coeff>}``

    Its package definition checks if the coefficient is ``1`` or ``-1``
    and then skips printing the ``1``, except for the coefficient of
    degree zero.  Also it sets the conditional deciding behaviour of
    :ref:`\\PolIfCoeffIsPlusOrMinusOne{T}{F} <PolIfCoeffIsPlusOrMinusOne>`.

    The actual printing of the coefficients, when not equal to plus or
    minus one, is handled by :ref:`\\PolTypesetOne{\<raw_coeff\>} <PolTypesetOne>`.

.. _PolIfCoeffIsPlusOrMinusOne:

``\PolIfCoeffIsPlusOrMinusOne{}{}``
***********************************

Syntax: ``\PolIfCoeffIsPlusOrMinusOne{T}{F}``

    This macro is a priori undefined.

    It is defined via the default :ref:`\\PolTypesetCmd{\<raw_coeff\>} <PolTypesetCmd>` to be
    used if needed in the execution of `\\PolTypesetMonomialCmd`_,
    e.g. to insert a ``\cdot`` in front of ``\PolVar^{\PolIndex}`` if
    the coefficient is not plus or minus one.

    The macro will execute ``T`` if the coefficient has been found to be
    plus or minus one, and ``F`` if not.  It chooses expandably between
    ``T`` and ``F``.

.. _PolTypesetOne:

``\PolTypesetOne{}``
^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolTypesetOne{<raw_coeff>}``

    Defaults to ``\xintTeXsignedFrac`` (LaTeX) or ``\xintTeXsignedOver``
    (else).  But these xintfrac_ old legacy macros are a bit
    annoying as they insist in exhibiting a power of ten rather than
    using simpler decimal notation.

    As alternative, one can do definitions such as::

      \def\PolTypesetOne#1{\xintDecToString{\xintREZ{#1}}}
      % or with LaTeX+siunitx for example
      \renewcommand\PolTypesetOne[1]{\num{\xintPFloat[5]{#1}}}
      % (as \num of siunitx understands floating point notation)
      \renewcommand\PolTypesetOne[1]{\num{\xintRound{4}{#1}}}

.. _PolTypesetMonomialCmd:

``\PolTypesetMonomialCmd``
^^^^^^^^^^^^^^^^^^^^^^^^^^

    This decides how a monomial (in variable ``\PolVar`` and with
    exponent ``\PolIndex``) is to be printed. The default does nothing
    for the constant term, ``\PolVar`` for the first degree and
    ``\PolVar^{\PolIndex}`` for higher degrees monomials. Beware that
    ``\PolIndex`` expands to digit tokens and needs termination in
    ``\ifnum`` tests.

.. _PolTypesetCmdPrefix:

``\PolTypesetCmdPrefix{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolTypesetCmdPrefix{<raw_coeff>}``

    Expands to a ``+`` if the ``raw_coeff`` is zero or positive, and to
    nothing if ``raw_coeff`` is negative, as in latter case the
    ``\xintTeXsignedFrac`` (or ``\xintTeXsignedOver``) used by
    :ref:`\\PolTypesetCmd{\<raw_coeff\>} <PolTypesetCmd>` will put the ``-`` sign in front of
    the fraction (if it is a fraction) and this will thus serve as
    separator in the typeset formula. Not used for the first term.


.. _PolTypeset*:

``\PolTypeset*[]{}``
~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolTypeset*[<letter>]{<pol. expr.>}``

    Typesets in ascending powers. The ``<letter>`` optional argument
    (after the ``*``) declares the letter to use in the *output*.
    As for `\\PolTypeset <PolTypeset>`_, it defaults to ``x``.

    To modify the expected ``x`` in the *input*, see `\\PolToExprInVar`_.

    Extended at ``0.8`` to accept general expressions and not only
    polynomial names.


.. _PolLet:

``\PolLet{}={}``
~~~~~~~~~~~~~~~~

Syntax: ``\PolLet{<polname_2>}={<polname_1>}``

    Makes a copy of the already defined polynomial ``polname_1`` to a
    new one ``polname_2``. This has the same effect as
    ``\PolDef{<polname_2>}{<polname_1>(x)}`` or (better)
    ``\PolDef{<polname_2>}{<polname_1>}`` but with less overhead. The
    ``=`` is optional.

.. _PolGlobalLet:

``\PolGlobalLet{}={}``
~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolGlobalLet{<polname_2>}={<polname_1>}``

    Acts globally.

.. _PolAssign:

``\PolAssign{}\toarray{}``
~~~~~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolAssign{<polname>}\toarray{<\macro>}``

    Defines a one-argument expandable macro ``\macro{#1}`` which expands
    to the (raw) #1th polynomial coefficient.

    - Attention, coefficients here are indexed starting at 1.  This is
      an unfortunate legacy situation related to the original indexing
      convention in xinttools_ arrays.

    - With #1=-1, -2, ..., ``\macro{#1}`` returns leading coefficients.

    - With #1=0, returns the number of coefficients, i.e. ``1 + deg f``
      for non-zero polynomials.

    - Out-of-range #1's return ``0/1[0]``.

    See also :ref:`\\PolNthCoeff{\<polname\>}{\<index\>} <PolNthCoeff>`.

.. _PolGet:

``\PolGet{}\fromarray{}``
~~~~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolGet{<polname>}\fromarray{<\macro>}``

    Does the converse operation to
    ``\PolAssign{<polname>}\toarray\macro``. Each individual
    ``\macro{<value>}`` gets expanded in an ``\edef`` and then normalized
    via xintfrac_\ 's macro ``\xintRaw``.

    The leading zeros are removed from the polynomial.

    (contrived) Example::

      \xintAssignArray{1}{-2}{5}{-3}\to\foo
      \PolGet{f}\fromarray\foo

    This will define ``f`` as would have ``\poldef f(x):=1-2x+5x^2-3x^3;``.

    .. vieux commentaire

       Prior to ``0.5``, coefficients were not normalized via
       ``\xintRaw`` for internal storage.

.. _PolFromCSV:

``\PolFromCSV{}{}``
~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolFromCSV{<polname>}{<csv>}``

    Defines a polynomial directly from the comma separated list of values
    (or a macro expanding to such a list) of its coefficients, the *first
    item* gives the constant term, the *last item* gives the leading
    coefficient, except if zero, then it is dropped (iteratively). List
    items are each expanded in an ``\edef`` and then put into normalized
    form via xintfrac_\ 's macro ``\xintRaw``.

    As leading zero coefficients are removed::

      \PolFromCSV{f}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

    defines the zero polynomial, which holds only one coefficient.

    See also expandable macro :ref:`\\PolToCSV{\<polname\>} <PolToCSV>`.

    .. vieux commentaire

       Prior to ``0.5``, coefficients were not normalized via
       ``\xintRaw`` for internal storage.


.. _PolMapCoeffs:

``\PolMapCoeffs{}{}``
~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolMapCoeffs{\macro}{<polname>}``

    It modifies ('in-place': original coefficients get lost) each
    coefficient of the defined polynomial via the *expandable* macro
    ``\macro``. The degree is adjusted as necessary if some leading
    coefficients vanish after the operation.

    In the replacement text of ``\macro``, ``\index`` expands to the
    coefficient index (starting at zero for the constant term).

    Notice that ``\macro`` will have to handle inputs in the xintfrac_
    internal format. This means that it probably will have to be
    expressed in terms of macros from the xintfrac_ package.

    Example::

      \def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}}

    (or with ``\xintSqr{\index}``) to replace ``n``-th coefficient
    ``f_n`` by ``f_n*n^2``.

.. _PolReduceCoeffs:

``\PolReduceCoeffs{}``
~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolReduceCoeffs{<polname>}``

    Reduces the internal representations of the coefficients to
    their lowest terms.

.. _PolReduceCoeffs*:

``\PolReduceCoeffs*{}``
~~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolReduceCoeffs*{<polname>}``

    Reduces the internal representations of the coefficients to their
    lowest terms, but ignoring a possible separated "power of ten part".

    For example, xintfrac_ stores an ``30e2/50`` input as ``30/50`` with
    a separate ``10^2`` part.  This will thus get replaced by ``3e^2/5``
    (or rather whatever xintfrac_ uses for internal representation), and
    not by ``60`` as would result from complete reduction.

    Evaluations with polynomials treated by this can be much faster than
    with those handled by the non-starred variant
    :ref:`\\PolReduceCoeffs{\<polname\>} <PolReduceCoeffs>`: as the numerators and denominators
    remain generally smaller.

.. _PolMakeMonic:

``\PolMakeMonic{}``
~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolMakeMonic{<polname>}``

    Divides by the leading coefficient. It is recommended to execute
    :ref:`\\PolReduceCoeffs*{\<polname\>} <PolReduceCoeffs*>` immediately afterwards. This is not
    done automatically, in case the original polynomial had integer
    coefficients and the user wants to keep the leading one as common
    denominator for typesetting purposes.

.. _PolMakePrimitive:

``\PolMakePrimitive{}``
~~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolMakePrimitive{<polname>}``

    Divides by the integer content see (`\\PolIContent`_).
    This thus produces a polynomial with integer
    coefficients having no common factor. The sign of the leading
    coefficient is not modified.

.. _PolDiff:

``\PolDiff{}{}``
~~~~~~~~~~~~~~~~

Syntax: ``\PolDiff{<polname_1>}{<polname_2>}``

    This sets ``polname_2`` to the first derivative of ``polname_1``. It
    is allowed to issue ``\PolDiff{f}{f}``, effectively replacing ``f``
    by ``f'``.

    Coefficients of the result ``polname_2`` are irreducible fractions
    (see `Technicalities`_ for the whole story.)

.. _PolDiff[]:

``\PolDiff[]{}{}``
~~~~~~~~~~~~~~~~~~

Syntax: ``\PolDiff[N]{<polname_1>}{<polname_2>}``

    This sets ``polname_2`` to the ``N``-th derivative of ``polname_1``.
    Identical arguments is allowed. With ``N=0``, same effect as
    ``\PolLet{<polname_2>}={<polname_1>}``. With negative ``N``, switches to
    using ``\PolAntiDiff``.

.. _PolAntiDiff:

``\PolAntiDiff{}{}``
~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolAntiDiff{<polname_1>}{<polname_2>}``

    This sets ``polname_2`` to the primitive of ``polname_1`` vanishing
    at zero.

    Coefficients of the result ``polname_2`` are irreducible fractions
    (see `Technicalities`_ for the whole story.)

.. _PolAntiDiff[]:

``\PolAntiDiff[]{}{}``
~~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolAntiDiff[N]{<polname_1>}{<polname_2>}``

    This sets ``polname_2`` to the result of ``N`` successive integrations on
    ``polname_1``. With negative ``N``, it switches to using ``\PolDiff``.

.. _PolDivide:

``\PolDivide{}{}{}{}``
~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolDivide{<polname_1>}{<polname_2>}{<polname_Q>}{<polname_R>}``

    This sets ``polname_Q`` and ``polname_R`` to be the quotient and
    remainder in the Euclidean division of ``polname_1`` by
    ``polname_2``.

.. _PolQuo:

``\PolQuo{}{}{}``
~~~~~~~~~~~~~~~~~

Syntax: ``\PolQuo{<polname_1>}{<polname_2>}{<polname_Q>}``

    This sets ``polname_Q`` to be the quotient in the Euclidean division
    of ``polname_1`` by ``polname_2``.

.. _PolRem:

``\PolRem{}{}{}``
~~~~~~~~~~~~~~~~~

Syntax: ``\PolRem{<polname_1>}{<polname_2>}{<polname_R>}``

    This sets ``polname_R`` to be the remainder in the Euclidean division
    of ``polname_1`` by ``polname_2``.

.. _PolGCD:

``\PolGCD{}{}{}``
~~~~~~~~~~~~~~~~~

Syntax: ``\PolGCD{<polname_1>}{<polname_2>}{<polname_GCD>}``

    This sets ``polname_GCD`` to be the (monic) GCD of ``polname_1``
    and ``polname_2``.  It is a unitary polynomial except if both
    ``polname_1`` and ``polname_2`` vanish, then ``polname_GCD`` is the
    zero polynomial.

.. ``\PolIGCD{<polname_1>}{<polname_2>}{polname_iGCD}``
   ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    **NOT YET**

    This **assumes** that the two polynomials have integer coefficients.
    It then computes the greatest common divisor in the integer
    polynomial ring, normalized to have a positive leading coefficient
    (if the inputs are not both zero).


Root localization routines via the `Sturm Theorem`_
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

As :ref:`\\PolToSturm{\<polname\>}{\<sturmname\>} <PolToSturm>` and
:ref:`\\PolSturmIsolateZeros{\<sturmname\>} <PolSturmIsolateZeros>` and variants declare
additional polynomial or scalar variables with names based on ``<sturmname>`` as
prefix, it is advisable to keep the ``<sturmname>`` namespace separate from
the one applying to ``\xintexpr`` variables generally, or to polynomials.

.. _PolToSturm:

``\PolToSturm{}{}``
^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolToSturm{<polname>}{<sturmname>}``

    With ``<polname>`` being for example ``P``, and ``<sturmname>`` being
    for example ``S``, the macro starts by computing the derivative
    ``P'``, then computes the opposite of the remainder in the euclidean
    division of ``P`` by ``P'``, then the opposite of the remainder in
    the euclidean division of ``P'`` by the first obtained polynomial,
    etc... Up to signs following the ``--++--++...`` pattern, these are
    the same remainders as in the Euclide algorithm applied to the
    computation of the GCD of ``P`` and ``P'``.

    The precise process differs from the above description: the
    algorithm first sets ``S_0_`` to be the *primitive part* of ``P``
    and ``S_1_`` to be the *primitive part* of ``P'`` (see
    :ref:`\\PolIContent{\<polname\>} <PolIContent>`), then at each step
    the remainder is made primitive and stored for internal reference as
    ``S_k_``, so only integer-coefficients polynomials are manipulated.

    .. warning::

       This exact procedure will perhaps in future be replaced by a
       *sub-resultant algorithm*, which may bring some speed gain in
       obtaining a pseudo-Sturm sequence, but some experimenting is
       needed, in the context of realistically realizable computations
       by the package; primitive polynomials although a bit costly
       have the smallest coefficients hence are the best for the kind of
       computations done for root localization, after having computed a
       Sturm sequence.

    The last non-zero primitivized remainder ``S_N_`` is, up to sign,
    the primitive part of the GCD of ``P`` and ``P'``.  Its roots (real
    and complex) are the multiple roots of the original ``P``.  The
    original ``P`` was "square-free" (i\.e\. did not have multiple real
    or complex roots) if and only if ``S_N_`` is a constant, which is
    then ``+1`` or ``-1`` (its value before primitivization is lost).
 
    The macro then divides each ``S_k_`` by ``S_N_`` and declares the
    quotients ``S_k`` as user polynomials for future use.  By Gauss
    theorem about the contents of integer-coefficients polynomials,
    these ``S_k`` also are primitive integer-coefficients polynomials.

    This step will be referred to as *normalization*, and in this
    documentation the obtained polynomials are said to constitute the
    "Sturm chain" (or "Sturm sequence"), i.e. by convention the "Sturm
    chain polynomials" are square-free and primitive.  The possibly
    non-square-free ones are referred to as *non-normalized*.

    As an exception to the rule, if the original ``P`` was "square-free"
    (i\.e\. did not have multiple real or complex roots) then
    normalization is skipped (in that case ``S_N_`` is either ``+1`` or
    ``-1``), so ``S_0_`` is exactly the primitive part of starting
    polynomial ``P``, in the "square-free" case.

    The next logical step is to execute `\\PolSturmIsolateZeros{S}
    <PolSturmIsolateZeros_>`_ or one of its variants.  Be careful not to
    use the names ``sturmname_0``, ``sturmname_1``, etc... for defining
    other polynomials after having done
    ``\PolToSturm{<polname>}{<sturmname>}`` and before executing
    ``\PolSturmIsolateZeros{<sturmname>}`` or its variants else the
    latter will behave erroneously.

    .. note::

       The declaration of the ``S_k``\ 's will overwrite with no warning
       previously declared polynomials with identical names ``S_k``,
       i.e. ``<sturmname>_k``.  This is why the macro was designed
       to expect two names: ``<polname>`` and ``<sturmname>``.

       It is allowed to use the polynomial name ``P`` as Sturm chain
       name ``S``: ``\PolToSturm{P}{P}``, but this is considered bad
       practice for the reason mentioned in the previous paragraph.

       Furthermore, `\\PolSturmIsolateZeros <PolSturmIsolateZeros_>`_
       creates xintexpr_ variables whose names start with
       ``<sturmname>L_``, ``<sturmname>R_``, and ``<sturmname>Z_``, also
       ``<sturmname>M_`` for holding the multiplicities, and this may
       overwrite pre-existing user-defined xintexpr_ variables.

    .. warning::

       The reason why the ``S_k``\ 's are declared as polynomials is
       that the associated polynomial functions are needed to compute
       the sign changes in the Sturm sequence evaluated at a given
       location, as this is the basis mechanism of `\\PolSturmIsolateZeros
       <PolSturmIsolateZeros_>`_ (on the basis of the `Sturm theorem`_).

       It is possible that in future the package will only internally
       construct such polynomial functions and only the starred variant
       will make the normalized (i.e. square-free) Sturm sequence public.

    The integer ``N`` giving the length of the Sturm chain ``S_0``,
    ``S_1``, ..., ``S_N`` is available as
    :ref:`\\PolSturmChainLength{\<sturmname\>} <PolSturmChainLength>`.  If all roots of original ``P``
    are real, then ``N`` is both the number of distinct real roots and
    the degree of ``S_0``.  In the case of existence of complex roots,
    the number of distinct real roots is at most ``N`` and ``N`` is at
    most the degree of ``S_0``.

.. _PolToSturm*:

``\PolToSturm*{}{}``
^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolToSturm*{<polname>}{<sturmname>}``

    Does the same as `un-starred version <PolToSturm_>`_ and additionally it
    keeps for user usage the memory of the *un-normalized* (but still
    made primitive) Sturm chain
    polynomials ``sturmname_k_``, ``k=0,1, ..., N``, with
    ``N`` being :ref:`\\PolSturmChainLength{\<sturmname\>} <PolSturmChainLength>`.

    .. comment

       The square-free part of ``<polname>`` is ``sturmname_0``, and their
       quotient is the polynomial with name
       ``sturmname_\PolSturmChainLength{<sturmname>}_``. It thus easy to
       set-up a loop iteratively computing the latter until the last one
       is a constant, thus obtaining the decomposition of an ``f`` as
       a product ``c f_1 f_2 f_3 ...`` of a constant and square-free (primitive)
       polynomials, where each ``f_i`` divides its predecessor.       

.. _PolSturmIsolateZeros:

``\PolSturmIsolateZeros{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmIsolateZeros{<sturmname>}``

    The macro locates, using the `Sturm Theorem`_, as many disjoint
    intervals as there are distinct real roots.

    .. important::

       The Sturm chain must have been produced by an earlier
       :ref:`\\PolToSturm{\<polname\>}{\<sturmname\>} <PolToSturm>`.
       
    After its execution they are two types of such intervals (stored in
    memory and accessible via macros or xintexpr_ variables, see below):

    - singleton ``{a}``: then ``a`` is a root, (necessarily a decimal
      number, but not all such decimal numbers are exactly identified yet).

    - open intervals ``(a,b)``: then there is exactly one root ``z``
      such that ``a < z < b``, and the end points are guaranteed to not
      be roots.

    The interval boundaries are decimal numbers, originating
    in iterated decimal subdivision from initial intervals
    ``(-10^E, 0)`` and ``(0, 10^E)`` with ``E`` chosen initially large
    enough so that all roots are enclosed; if zero is a root it is always
    identified as such. The non-singleton intervals are of the
    type ``(a/10^f, (a+1)/10^f)`` with ``a`` an integer, which is
    neither ``0`` nor ``-1``. Hence either ``a`` and ``a+1`` are both positive
    or they are both negative.

    One does not *a priori* know what will be the lengths of these
    intervals (except that they are always powers of ten), they
    vary depending on how many digits two successive roots have in
    common in their respective decimal expansions.

    .. important::

       If some two consecutive intervals share an end-point, no
       information is yet gained about the separation between the two
       roots which could at this stage be arbitrarily small.

       See :ref:`\\PolRefineInterval*{\<sturmname\>}{\<index\>} <PolRefineInterval*>` which addresses
       this issue.

    .. This procedure is covariant
       with the independent variable ``x`` becoming ``-x``.
       Hmm, pas sûr et trop fatigué

    Let us suppose ``<sturmname>`` is ``S``.

    The interval boundaries (and exactly found roots) are made available
    for future computations in ``\xintexpr/xinteval`` or ``\poldef`` as
    variables ``SL_1``, ``SL_2``, etc..., for the left end-points and
    ``SR_1``, ``SR_2``, ..., for the right end-points.

    Additionally, xintexpr_ variable ``SZ_1_isknown`` will have value
    ``1`` if the root in the first interval is known, and ``0``
    otherwise. And similarly for the other intervals.

    .. important::

       The variable declarations are done with no check of existence of
       previously existing variables with identical names.

    Also, macros :ref:`\\PolSturmIsolatedZeroLeft{\<sturmname\>}{\<index\>} <PolSturmIsolatedZeroLeft>` and
    :ref:`\\PolSturmIsolatedZeroRight{\<sturmname\>}{\<index\>} <PolSturmIsolatedZeroRight>` are provided which
    expand to these same values, written in decimal notation (i.e.
    pre-processed by `\\PolDecToString <PolDecToString_>`_.) And there
    is also :ref:`\\PolSturmIfZeroExactlyKnown{\<sturmname\>}{\<index\>}{T}{F} <PolSturmIfZeroExactlyKnown>`.

    .. important::

       Trailing zeroes in the stored decimal numbers accessible via the
       macros are significant: they are also present in the decimal
       expansion of the exact root, so as to be able for example to
       print out bounds of real roots with as many digits as is
       significant, even if the digits are zeros.

    The start of the decimal expansion of the ``<index>``-th root is given by
    `\\PolSturmIsolatedZeroLeft{<sturmname>}{<index>}
    <PolSturmIsolatedZeroLeft_>`_ if the root is positive, and by
    `\PolSturmIsolatedZeroRight{<sturmname>}{<index>}
    <PolSturmIsolatedZeroRight_>`_ if the root is neagtive. These two
    decimal numbers are either both zero or both of the same sign.

    The number of distinct roots is obtainable expandably as
    :ref:`\\PolSturmNbOfIsolatedZeros{\<sturmname\>} <PolSturmNbOfIsolatedZeros>`.

    Furthermore
    :ref:`\\PolSturmNbOfRootsOf{\<sturmname\>}\\LessThanOrEqualTo{\<value\>} <PolSturmNbOfRootsOfLessThanOrEqualTo>` and
    :ref:`\\PolSturmNbOfRootsOf{\<sturmname\>}\\LessThanOrEqualToExpr{\<num. expr.\>} <PolSturmNbOfRootsOfLessThanOrEqualToExpr>`.
    will expandably compute respectively the number of real roots at
    most equal to ``value`` or ``expression``, and the same but with
    multiplicities.

    These variables and macros are automatically updated in case of
    subsequent usage of :ref:`\\PolRefineInterval*{\<sturmname\>}{\<index\>} <PolRefineInterval*>` or
    other localization improving macros.

    .. note::

       The current polexpr implementation defines the xintexpr_ variables
       and xinttools_ arrays as described above with global scope. On the
       other hand the Sturm sequence polynomials obey the current scope.

       This is perhaps a bit inconsistent and may change in future.

    .. note::

       The results are exact
       bounds for the mathematically exact real roots.

       Future releases will perhaps also provide macros based on Newton
       or Regula Falsi methods. Exact computations with such methods
       lead however quickly to very big fractions, and this forces usage
       of some rounding scheme for the abscissas if computation times
       are to remain reasonable. This raises issues of its own, which
       are studied in numerical mathematics.

.. _PolSturmIsolateZeros*:

``\PolSturmIsolateZeros*{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmIsolateZeros*{<sturmname>}``

    The macro does the same as :ref:`\\PolSturmIsolateZeros{\<sturmname\>} <PolSturmIsolateZeros>` and
    then in addition it does the extra work to determine all
    multiplicities of the real roots.

    After execution,
    :ref:`\\PolSturmIsolatedZeroMultiplicity{\<sturmname\>}{\<index\>} <PolSturmIsolatedZeroMultiplicity>` expands
    to the multiplicity of the root located in the ``index``\ -th
    interval (intervals are enumerated from left to right, with index
    starting at ``1``).

    Furthermore, if for example the ``<sturmname>`` is ``S``, xintexpr_
    variables ``SM_1``, ``SM_2``... hold the multiplicities thus
    computed.

    .. note::

       Somewhat counter-intuitively, it is not necessary to have
       executed the :ref:`\\PolToSturm* <PolToSturm*>`
       starred variant: during its
       execution, :ref:`\\PolToSturm <PolToSturm>`,
       even though it does not declare the
       non-square-free Sturm chain polynomials as user-level genuine
       polynomials, stores their data in private macros.

    See ``The degree nine polynomial with 0.99, 0.999, 0.9999 as triple
    roots`` example in ``polexpr-examples.pdf``.

.. _PolSturmIsolateZerosAndGetMultiplicities:

``\PolSturmIsolateZerosAndGetMultiplicities{}``
***********************************************

Syntax: ``\PolSturmIsolateZerosAndGetMultiplicities{<sturmname>}``

    This is another name for :ref:`\\PolSturmIsolateZeros*{\<sturmname\>} <PolSturmIsolateZeros*>`.

.. _PolSturmIsolateZeros**:

``\PolSturmIsolateZeros**{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmIsolateZeros**{<sturmname>}``

    The macro does the same as :ref:`\\PolSturmIsolateZeros*{\<sturmname\>} <PolSturmIsolateZeros*>` and
    in addition it does the extra work to determine all the *rational*
    roots.

    .. note::

       After execution of this macro, a root is "known" if and only if
       it is rational.

    Furthermore, primitive polynomial ``sturmname_sqf_norr`` is created
    to match the (square-free) ``sturmname_0`` from which all rational
    roots have been removed. The number of distinct rational roots is
    thus the difference between the degrees of these two polynomials
    (see also :ref:`\\PolSturmNbOfRationalRoots{\<sturmname\>}
    <PolSturmNbOfRationalRoots>`).

    And ``sturmname_norr`` is ``sturmname_0_`` from which all rational
    roots have been removed, i.e. it contains the irrational roots of
    the original polynomial, with the same multiplicities.

    See ``A degree five polynomial with three rational
    roots`` in ``polexpr-examples.pdf``.

.. _PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots:

``\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots``
**********************************************************

Syntax: ``\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots``

    This is another name for :ref:`\\PolSturmIsolateZeros**{\<sturmname\>} <PolSturmIsolateZeros**>`.

.. _PolSturmIsolateZerosAndFindRationalRoots:

``\PolSturmIsolateZerosAndFindRationalRoots{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmIsolateZerosAndFindRationalRoots{<sturmname>}``

    This works exactly like :ref:`\\PolSturmIsolateZeros**{\<sturmname\>} <PolSturmIsolateZeros**>`
    (inclusive of declaring the polynomials ``sturmname_sqf_norr`` and
    ``sturmname_norr`` with no rational roots) except that it does *not*
    compute the multiplicities of the *non-rational* roots.

    .. note::

       There is no macro to find the rational roots but not compute
       their multiplicities at the same time.

    .. attention::

       This macro does *not* define xintexpr_ variables
       ``sturmnameM_1``, ``sturmnameM_2``, ... holding the
       multiplicities and it leaves the multiplicity array (whose accessor
       is :ref:`\\PolSturmIsolatedZeroMultiplicity{\<sturmname\>}{\<index\>} <PolSturmIsolatedZeroMultiplicity>`) into
       a broken state, as all non-rational roots will supposedly have
       multiplicity one. This means that the output of
       `\\PolPrintIntervals* <PolPrintIntervals*_>`_ will be
       erroneous regarding the multiplicities of irrational roots.

       I decided to document it because finding multiplicities of the
       non rational roots is somewhat costly, and one may be interested
       only into finding the rational roots (of course random
       polynomials with integer coefficients will not have *any*
       rational root anyhow).


.. _PolRefineInterval*:

``\PolRefineInterval*{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolRefineInterval*{<sturmname>}{<index>}``

    The ``index``\ -th interval (starting indexing at one) is further
    subdivided as many times as is necessary in order for the newer
    interval to have both its end-points distinct from the end-points of
    the original interval.  As a consequence, the ``k``\ th root is then
    strictly separated from the other roots.

.. _PolRefineInterval[]:

``\PolRefineInterval[]{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolRefineInterval[N]{<sturmname>}{<index>}``

    The ``index``\ -th interval (starting count at one) is further
    subdivided once, reducing its length by a factor of 10. This is done
    ``N`` times if the optional argument ``[N]`` is present.

.. _PolEnsureIntervalLength:

``\PolEnsureIntervalLength{}{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolEnsureIntervalLength{<sturmname>}{<index>}{<exponent>}``

    The ``index``\ -th interval is subdivided until its length becomes at
    most ``10^E``. This means (for ``E<0``) that the first ``-E`` digits
    after decimal mark of the ``k``\ th root will then be known exactly.

.. _PolEnsureIntervalLengths:

``\PolEnsureIntervalLengths{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolEnsureIntervalLengths{<sturmname>}{<exponent>}``

    The intervals as obtained from ``\PolSturmIsolateZeros`` are (if
    necessary) subdivided further by (base 10) dichotomy in order for
    each of them to have length at most ``10^E``.

    This means that decimal expansions of all roots will be known with
    ``-E`` digits (for ``E<0``) after decimal mark.

.. _PolSetToSturmChainSignChangesAt:

``\PolSetToSturmChainSignChangesAt{}{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSetToSturmChainSignChangesAt{\foo}{<sturmname>}{<value>}``

    Sets macro ``\foo`` to store the number of sign changes in the
    already computed normalized Sturm chain with name prefix
    ``<sturmname>``, at location ``<value>`` (which must be in format as
    acceptable by the xintfrac_ macros.)

    The definition is made with global scope. For local scope, use
    ``[\empty]`` as extra optional argument.

    One can use this immediately after creation of the Sturm chain.

.. _PolSetToNbOfZerosWithin:

``\PolSetToNbOfZerosWithin{}{}{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSetToNbOfZerosWithin{\foo}{<sturmname>}{<value_left>}{<value_right>}``

    Sets, assuming the normalized Sturm chain has been already computed,
    macro ``\foo`` to store the number of roots of ``sturmname_0`` in
    the interval ``(value_left, value_right]``.  The macro first
    re-orders end-points if necessary for ``value_left <= value_right``
    to hold.

    In accordance to `Sturm Theorem`_ this is computed as the difference
    between the number of sign changes of the Sturm chain at ``value_right``
    and the one at ``value_left``.

    The definition is made with global scope. For local scope, use
    ``[\empty]`` as extra optional argument.

    One can use this immediately after creation of a Sturm chain.

    See also the expandable
    :ref:`\\PolSturmNbOfRootsOf{\<sturmname\>}\\LessThanOrEqualTo{value} <PolSturmNbOfRootsOfLessThanOrEqualTo>`,
    which however requires prior execution of `\\PolSturmIsolateZeros
    <PolSturmIsolateZeros_>`_.

    See also the expandable
    :ref:`\\PolSturmNbWithMultOfRootsOf{\<sturmname\>}\\LessThanOrEqualTo{value} <PolSturmNbWithMultOfRootsOfLessThanOrEqualTo>`
    which requires prior execution of
    `\\PolSturmIsolateZeros* <PolSturmIsolateZeros*_>`_.

.. _PolPrintIntervals:

Displaying the found roots: ``\PolPrintIntervals[<varname>]{}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolPrintIntervals[<varname>]{<sturmname>}``

    This is a convenience macro which prints the bounds for the roots
    ``Z_1``, ``Z_2``, ... (the optional argument ``varname`` allows to
    specify a replacement for the default ``Z``). This will be done (by
    default) in a
    math mode ``array``, one interval per row, and pattern ``rcccl``,
    where the second and fourth column hold the ``<`` sign, except when
    the interval reduces to a singleton, which means the root is known
    exactly.

    .. note::

       The explanations here and in this section are for LaTeX.  With
       other TeX macro formats, the LaTeX syntax such as for example
       ``\begin{array}{rcccl}`` which appears in the documentation here
       is actually replaced with quasi-equivalent direct use of TeX
       primitives.

    The next macros which govern its output.

.. _PolPrintIntervalsNoRealRoots:

``\PolPrintIntervalsNoRealRoots``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Executed in place of an ``array`` environment, when there are no
    real roots. Default definition::

      \newcommand\PolPrintIntervalsNoRealRoots{}

.. _PolPrintIntervalsBeginEnv:

``\PolPrintIntervalsBeginEnv``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Default definition (given here for LaTeX, Plain has a variant)::

      \newcommand\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}}

    A simpler ``center`` environment provides a straightforward way to
    obtain a display allowing pagebreaks. Of course redefinitions must
    at any rate be kept in sync with `\\PolPrintIntervalsKnownRoot`_ and
    `\\PolPrintIntervalsUnknownRoot`_.

    Prior to ``0.8.6`` it was not possible to use here for example
    ``\begin{align}`` due to the latter executing twice in contents.


.. _PolPrintIntervalsEndEnv:

``\PolPrintIntervalsEndEnv``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Default definition::

      \newcommand\PolPrintIntervalsEndEnv{\end{array}\]}

.. _PolPrintIntervalsRowSeparator:

``\PolPrintIntervalsRowSeparator``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Expands by default to ``\\`` with LaTeX and to  ``\cr`` with Plain

    Added at ``0.8.6``.

.. _PolPrintIntervalsKnownRoot:

``\PolPrintIntervalsKnownRoot``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Default definition::

      \newcommand\PolPrintIntervalsKnownRoot{%
        &&\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}%
        &=&\PolPrintIntervalsPrintExactZero
      }

.. _PolPrintIntervalsUnknownRoot:

``\PolPrintIntervalsUnknownRoot``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Default definition::

      \newcommand\PolPrintIntervalsUnknownRoot{%
        \PolPrintIntervalsPrintLeftEndPoint&<&%
        \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&<&%
        \PolPrintIntervalsPrintRightEndPoint
      }


.. _PolPrintIntervalsPrintExactZero:

``\PolPrintIntervalsPrintExactZero``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Default definition::

      \newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheLeftEndPoint}


.. _PolPrintIntervalsPrintLeftEndPoint:

``\PolPrintIntervalsPrintLeftEndPoint``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Default definition::

      \newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheLeftEndPoint}

.. _PolPrintIntervalsPrintRightEndPoint:

``\PolPrintIntervalsPrintRightEndPoint``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Default definition is::

      \newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint}

.. _PolPrintIntervals*:

``\PolPrintIntervals*[<varname>]{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolPrintIntervals*[<varname>]{<sturmname>}``

    This starred variant produces an alternative output (which
    displays the root multiplicity), and is provided as an
    example of customization.

    As replacement for `\\PolPrintIntervalsKnownRoot`_,
    `\\PolPrintIntervalsPrintExactZero`_,
    `\\PolPrintIntervalsUnknownRoot`_ it uses its own
    ``\POL@@PrintIntervals...`` macros. We only reproduce here one
    definition::

      \newcommand\POL@@PrintIntervalsPrintExactZero{%
         \displaystyle
         \xintTeXsignedFrac{\PolPrintIntervalsTheLeftEndPoint}%
      }%
    
    Multiplicities are printed using this auxiliary macro:

.. _PolPrintIntervalsPrintMultiplicity:

``\PolPrintIntervalsPrintMultiplicity``
***************************************

    whose default definition is::

      \newcommand\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)}


Expandable macros
-----------------

.. note::

   At ``0.8`` ``polexpr`` is usable with Plain TeX and not only with
   LaTeX.  Some examples given in this section may be using LaTeX syntax
   such as ``\renewcommand``.  Convert to TeX primitives as appropriate
   if testing with a non LaTeX macro format.

These macros expand completely in two steps except ``\PolToExpr`` and
``\PolToFloatExpr`` which need a ``\write``, ``\edef`` or a
``\csname...\endcsname`` context.

.. _PolToExpr:

``\PolToExpr{}``
~~~~~~~~~~~~~~~~

Syntax: ``\PolToExpr{<pol. expr.>}``

    Produces expandably [#]_ the string ``coeff_N*x^N+...``, i.e. the
    polynomial is using descending powers.

    .. [#] requires exhaustive expansion, for example as triggered by
           ``\write`` or ``\edef``.

    Since ``0.8`` the input is not restricted to be a polynomial name but
    is allowed to be an arbitrary expression.  Then ``x`` is expected as
    indeterminate but this can be customized via `\\PolToExprInVar`_.

    The output uses the letter ``x`` by default, this is customizable
    via `\\PolToExprVar`_.  The default output is compatible both with

    - the Maple's input format,

    - and the PSTricks ``\psplot[algebraic]`` input format.

    Attention that it is not compatible with Python, see further
    `\\PolToExprCaret`_ in this context.

    The following applies:

    - vanishing coefficients are skipped (issue ``\poltoexpralltrue`` to
      override this and produce output such as ``x^3+0*x^2+0*x^1+0``),

    - negative coefficients are not prefixed by  a ``+`` sign (else,
      Maple would not be happy),

    - coefficients numerically equal to ``1`` (or ``-1``) are present
      only via their sign,

    - the letter ``x`` is used and the degree one monomial is output as 
      ``x``, not as ``x^1``.

    - (``0.8``) the caret ``^`` is of catcode 12.  This means that one
      can for convenience typeset in regular text mode, for example
      using ``\texttt`` (in LaTeX). But TeX will not know how to break
      the expression across end-of-lines anyhow.  Formerly ``^`` was
      suitable for math mode but as the exponent is not braced this
      worked only for polynomials of degrees at most 9.  Anyhow this
      is not supposed to be a typesetting macro.

    Complete customization is possible, see the next macros.  Any user
    redefinition must maintain the expandability property.

.. _PolToExprVar:

``\PolToExprVar``
^^^^^^^^^^^^^^^^^

    Defaults to ``x``. The letter used in the macro output.


.. _PolToExprInVar:

``\PolToExprInVar``
^^^^^^^^^^^^^^^^^^^

    Defaults to ``x``: the letter used as the polynomial indeterminate
    in the macro input::

      \def\PolToExprInVar{x}% (default)

    Recall that declared polynomials are more efficiently used in
    algebraic expressions without the ``(x)``, i.e. ``P*Q`` is better
    than ``P(x)*Q(x)``.  Thus the input, even if an expression, does not
    have to contain any ``x``.

    (new with ``0.8``)

.. _PolToExprTimes:

``\PolToExprTimes``
^^^^^^^^^^^^^^^^^^^

    Defaults to ``*``.

.. _PolToExprCaret:

``\PolToExprCaret``
^^^^^^^^^^^^^^^^^^^

    Defaults to ``^`` of catcode 12.  Set it to
    expand to  ``**`` for Python compatible output.

    (new with ``0.8``) 

.. _PolToExprCmd:

``\PolToExprCmd{}``
^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolToExprCmd{<raw_coeff>}``

    Defaults to ``\xintPRaw{\xintRawWithZeros{#1}}``.

    This means that the coefficient value is printed-out as a fraction
    ``a/b``, skipping the ``/b`` part if ``b`` turns out to be one.

    Configure it to be ``\xintPRaw{\xintIrr{#1}}`` if the fractions
    must be in irreducible terms.

    An alternative is ``\xintDecToString{\xintREZ{#1}}`` which uses
    integer or decimal fixed point format such as ``23.0071`` if the
    internal representation of the number only has a power of ten as
    denominator (the effect of ``\xintREZ`` here is to remove trailing
    decimal zeros).  The behaviour of ``\xintDecToString`` is not yet
    stable for other cases, and for example at time of writing no
    attempt is made to identify inputs having a finite decimal expansion
    so for example ``23.007/2`` or ``23.007/25`` can appear in output
    and not their finite decimal expansion with no denominator.

.. _PolToExprOneTerm:

``\PolToExprOneTerm{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolToExprOneTerm{<raw_coeff>}{<exponent>}``

    This is the macro which from the coefficient and the exponent
    produces the corresponding term in output, such as ``2/3*x^7``.

    For its default definition, see the source code.  It uses
    `\\PolToExprCmd <PolToExprCmd_>`_, `\\PolToExprTimes
    <PolToExprTimes_>`_, `\\PolToExprVar <PolToExprVar_>`_ and
    `\\PolToExprCaret`_. 

.. _PolToExprOneTermStyleA:

``\PolToExprOneTermStyleA{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolToExprOneTermStyleA{<raw_coeff>}{<exponent>}``

    This holds the default package meaning of ``\PolToExprOneTerm``.

.. _PolToExprOneTermStyleB:

``\PolToExprOneTermStyleB{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolToExprOneTermStyleB{<raw_coeff>}{<exponent>}``

    This holds an alternative meaning, which puts the fractional part of
    a coefficient after the monomial, i.e. like this::

      2*x^11/3+3*x^8/7-x^5-x^4/4-x^3-x^2/2-2*x+1

    `\\PolToExprCmd <PolToExprCmd_>`_ isn't used at all in this style.  But
    `\\PolToExprTimes <PolToExprTimes_>`_, `\\PolToExprVar
    <PolToExprVar_>`_ and `\\PolToExprCaret`_ are obeyed.

    To activate it use ``\let\PolToExprOneTerm\PolToExprOneTermStyleB``.
    To revert to the package default behaviour, issue
    ``\let\PolToExprOneTerm\PolToExprOneTermStyleA``.

.. _PolToExprTermPrefix:

``\PolToExprTermPrefix{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolToExprTermPrefix{<raw_coeff>}``

    It receives as argument the coefficient.  Its default behaviour is
    to produce a ``+`` if the coefficient is positive, which will thus
    serve to separate the monomials in the output.  This is to match
    the default for :ref:`\\PolToExprCmd{\<raw_coeff\>} <PolToExprCmd>` which in case of a
    positive coefficient does not output an explicit ``+`` prefix.

.. _PolToFloatExpr:

``\PolToFloatExpr{}``
~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolToFloatExpr{<pol. expr.>}``

    Similar to :ref:`\\PolToExpr{\<pol. expr.\>} <PolToExpr>` but using
    :ref:`\\PolToFloatExprCmd{\<raw_coeff\>} <PolToFloatExprCmd>` which by default rounds and
    converts the coefficients to floating point format.

    .. note::

       This is unrelated to :ref:`\\PolGenFloatVariant{\<polname\>} <PolGenFloatVariant>`:
       :ref:`\\PolToFloatExprCmd{\<raw_coeff\>} <PolToFloatExprCmd>` operates on the *exact*
       coefficients anew (and may thus produce something else than
       the coefficients of the polynomial function acting
       in ``\xintfloateval`` if the floating point precision was changed
       in between).

    Extended at ``0.8`` to accept general expressions as input.

.. _PolToFloatExprOneTerm:

``\PolToFloatExprOneTerm{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolToFloatExprOneTerm{<raw_coeff>}{<exponent>}``

    Similar to :ref:`\\PolToExprOneTerm{\<raw_coeff\>}{\<exponent\>} <PolToExprOneTerm>`. But does not treat
    especially coefficients equal to plus or minus one.

.. _PolToFloatExprCmd:

``\PolToFloatExprCmd{}``
^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolToFloatExprCmd{<raw_coeff>}``

    The one-argument macro used by ``\PolToFloatExprOneTerm``.
    It defaults to ``\xintPFloat{#1}``, which trims trailing
    zeroes.

    *changed at 0.8.2* Formerly it was using ``\xintFloat``.

.. _PolToExpr*:

``\PolToExpr*{}``
~~~~~~~~~~~~~~~~~

Syntax: ``\PolToExpr*{<pol. expr.>}``

    Ascending powers: ``coeff_0+coeff_1*x+coeff_2*x^2+...``.

    Extended at ``0.8`` to accept general expressions as input.

    Customizable with the same macros as for
    :ref:`\\PolToExpr{\<pol. expr.\>} <PolToExpr>`.

.. _PolToFloatExpr*:

``\PolToFloatExpr*{}``
~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolToFloatExpr*{<pol. expr.>}``

    Ascending powers.

    Extended at ``0.8`` to accept general expressions as input.

.. _PolNthCoeff:

``\PolNthCoeff{}{}``
~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolNthCoeff{<polname>}{<index>}``

    It expands to the raw ``N``-th coefficient (``N=0`` corresponds to
    the constant coefficient).  If ``N`` is out of range, zero (in its
    default xintfrac_ format ``0/1[0]``) is returned.

    Negative indices ``N=-1``, ``-2``, ... return the leading
    coefficient, sub-leading coefficient, ..., and finally ``0/1[0]``
    for ``N<-1-degree``.

.. _PolLeadingCoeff:

``\PolLeadingCoeff{}``
~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolLeadingCoeff{<polname>}``

    Expands to the leading coefficient.

.. _PolDegree:

``\PolDegree{}``
~~~~~~~~~~~~~~~~

Syntax: ``\PolDegree{<polname>}``

    It expands to the degree. This is ``-1`` if zero polynomial but this
    may change in future. Should it then expand to ``-\infty`` ?

.. _PolIContent:

``\PolIContent{}``
~~~~~~~~~~~~~~~~~~

Syntax: ``\PolIContent{<polname>}``

    It expands to the contents of the polynomial, i.e. to the positive
    fraction such that dividing by this fraction produces a polynomial
    with integer coefficients having no common prime divisor.

    See `\\PolMakePrimitive <PolMakePrimitive_>`_.

.. _PolToList:

``\PolToList{}``
~~~~~~~~~~~~~~~~

Syntax: ``\PolToList{<polname>}``

    Expands to ``{coeff_0}{coeff_1}...{coeff_N}`` with ``N`` = degree, and
    ``coeff_N`` the leading coefficient
    (the zero polynomial does give ``{0/1[0]}`` and not an
    empty output.)

.. _PolToCSV:

``\PolToCSV{}``
~~~~~~~~~~~~~~~

Syntax: ``\PolToCSV{<polname>}``

    Expands to ``coeff_0, coeff_1, coeff_2, ....., coeff_N``, starting
    with constant term and ending with leading coefficient. Converse
    to :ref:`\\PolFromCSV{\<polname\>}{\<csv\>} <PolFromCSV>`.


.. _PolEvalAtExpr:

``\PolEval{}\AtExpr{}``
~~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolEval{<polname>}\AtExpr{<num. expr.>}``

    Same output as
    ``\xinteval{polname(numerical expression)}``.

.. _PolEvalAt:

``\PolEval{}\At{}``
~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolEval{<polname>}\At{<value>}``

    Evaluates the polynomial at the given value which must be in (or
    expand to) a format acceptable to the xintfrac_ macros.

.. _PolEvalReducedAtExpr:

``\PolEvalReduced{}\AtExpr{}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolEvalReduced{<polname>}\AtExpr{<num. expr.>}``

    Same output as ``\xinteval{reduce(polname(numerical expression))}``.

.. _PolEvalReducedAt:

``\PolEvalReduced{}\At{}``
~~~~~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolEvalReduced{<polname>}\At{<value>}``

    Evaluates the polynomial at the value which must be in (or expand
    to) a format acceptable to the xintfrac_ macros, and outputs an
    irreducible fraction.

.. _PolFloatEvalAtExpr:

``\PolFloatEval{}\AtExpr{}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolFloatEval{<polname>}\AtExpr{<num. expr.>}``

    Same output as ``\xintfloateval{polname(numerical expression)}``.

    .. attention::

       `\\PolGenFloatVariant <PolGenFloatVariant_>`_ must have
       been issued before.

    To use the *exact coefficients* with *exactly executed* additions
    and multiplications and do the rounding only as the final last step,
    the following syntax can be used: [#]_

    ::

        \xintfloateval{3.27*\xintexpr f(2.53)\relax^2}


    .. [#] Cf. xintexpr_ documentation about nested expressions.

.. _PolFloatEvalAt:

``\PolFloatEval{}\At{}``
~~~~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolFloatEval{<polname>}\At{<value>}``

    Evaluates the polynomial at the value which must be in (or expand
    to) a format acceptable to the xintfrac_ macros.


Expandable macros in relation to root localization via `Sturm Theorem`_
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. _PolSturmChainLength:

``\PolSturmChainLength{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmChainLength{<sturmname>}``

    Returns the integer ``N`` such that ``sturmname_N`` is the last one
    in the Sturm chain ``sturmname_0``, ``sturmname_1``, ...

    See :ref:`\\PolToSturm{\<polname\>}{\<sturmname\>} <PolToSturm>`.

.. _PolSturmIfZeroExactlyKnown:

``\PolSturmIfZeroExactlyKnown{}{}{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmIfZeroExactlyKnown{<sturmname>}{<index>}{T}{F}``

    Executes ``T`` if the ``index``\ -th interval reduces to a singleton,
    i.e. the root is known exactly, else ``F``.

    .. vieux commentaire

       ``index`` is allowed to be something like ``1+2*3`` as it is fed
       to ``\the\numexpr...\relax``.

.. _PolSturmIsolatedZeroLeft:

``\PolSturmIsolatedZeroLeft{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmIsolatedZeroLeft{<sturmname>}{<index>}``

    Expands to the left end-point for the ``index``\ -th interval, as
    computed by some earlier :ref:`\\PolSturmIsolateZeros{\<sturmname\>} <PolSturmIsolateZeros>`.

    .. note::

       Execution of this macro after some
       `\\PolRefineInterval{<sturmname>}{<index>} <PolRefineInterval[]_>`_
       will take into account the now known tighter bounds.

    The value is pre-formatted using `\\PolDecTostring
    <PolDecToString_>`_.

.. _PolSturmIsolatedZeroRight:

``\PolSturmIsolatedZeroRight{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmIsolatedZeroRight{<sturmname>}{<index>}``

    Expands to the right end-point for the ``index``\ -th interval as
    computed by some earlier :ref:`\\PolSturmIsolateZeros{\<sturmname\>} <PolSturmIsolateZeros>` and
    possibly refined afterwards.

    The value is pre-formatted using `\\PolDecTostring
    <PolDecToString_>`_.

.. _PolSturmIsolatedZeroMultiplicity:

``\PolSturmIsolatedZeroMultiplicity{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmIsolatedZeroMultiplicity{<sturmname>}{<index>}``

    Expands to the multiplicity of the unique root contained in the
    ``index``\ -th interval.

    .. attention::

       A prior execution of :ref:`\\PolSturmIsolateZeros*{\<sturmname\>} <PolSturmIsolateZeros*>` is mandatory.

    See ``The degree nine polynomial with 0.99, 0.999, 0.9999 as triple
    roots`` in ``polexpr-examples.pdf``.

.. _PolSturmNbOfIsolatedZeros:

``\PolSturmNbOfIsolatedZeros{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmNbOfIsolatedZeros{<sturmname>}``

    Expands to the number of real roots of the polynomial
    ``<sturmname>_0``, i.e. the number of distinct real roots of the
    polynomial originally used to create the Sturm chain via
    :ref:`\\PolToSturm{\<polname\>}{\<sturmname\>} <PolToSturm>`.

.. warning::

   The next few macros counting roots, with or without multiplicities,
   less than or equal to some value, are under evaluation and may be
   removed from the package if their utility is judged to be not high
   enough. They can be re-coded at user level on the basis of the other
   documented package macros anyway.

.. _PolSturmNbOfRootsOfLessThanOrEqualTo:

``\PolSturmNbOfRootsOf{}\LessThanOrEqualTo{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmNbOfRootsOf{<sturmname>}\LessThanOrEqualTo{<value>}``

    Expands to the number of distinct roots (of the polynomial used to
    create the Sturm chain) less than or equal to the ``value`` (i.e. a
    number of fraction recognizable by the xintfrac_ macros).

    .. attention::

       :ref:`\\PolSturmIsolateZeros{\<sturmname\>} <PolSturmIsolateZeros>` must have been executed
       beforehand.

       And the argument is a ``<sturmname>``, not a ``<polname>`` (this is
       why the macro contains Sturm in its name), simply to be reminded
       of the above constraint.

.. _PolSturmNbOfRootsOfLessThanOrEqualToExpr:

``\PolSturmNbOfRootsOf{}\LessThanOrEqualToExpr{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmNbOfRootsOf{<sturmname>}\LessThanOrEqualToExpr{<num. expr.>}``

    Expands to the number of distinct roots (of the polynomial 
    used to create the Sturm chain) which are less than or equal to the
    given numerical expression.

    .. attention::

       :ref:`\\PolSturmIsolateZeros{\<sturmname\>} <PolSturmIsolateZeros>` must have been executed
       beforehand.

.. _PolSturmNbWithMultOfRootsOfLessThanOrEqualTo:

``\PolSturmNbWithMultOfRootsOf{}\LessThanOrEqualTo{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmNbWithMultOfRootsOf{<sturmname>}\LessThanOrEqualTo{<value>}``

    Expands to the number counted with multiplicities of the roots (of
    the polynomial used to create the Sturm chain) which are less than
    or equal to the given ``value``.

    .. attention::

       :ref:`\\PolSturmIsolateZeros*{\<sturmname\>} <PolSturmIsolateZeros*>` (or the double starred
       variant) must have been executed beforehand.

.. _PolSturmNbWithMultOfRootsOfLessThanOrEqualToExpr:

``\PolSturmNbWithMultOfRootsOf{}\LessThanOrEqualToExpr{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmNbWithMultOfRootsOf{<sturmname>}\LessThanOrEqualToExpr{<num. expr.>}``

    Expands to the total number of roots (counted with multiplicities)
    which are less than or equal to the given ``expression``.

    .. attention::

       :ref:`\\PolSturmIsolateZeros*{\<sturmname\>} <PolSturmIsolateZeros*>` (or the double starred
       variant) must have been executed beforehand.

.. _PolSturmNbOfRationalRoots:

``\PolSturmNbOfRationalRoots{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmNbOfRationalRoots{<sturmname>}``

    Expands to the number of rational roots (without multiplicities).

    .. attention::

       :ref:`\\PolSturmIsolateZeros**{\<sturmname\>} <PolSturmIsolateZeros**>` must have been executed
       beforehand.

.. _PolSturmNbOfRationalRootsWithMultiplicities:

``\PolSturmNbOfRationalRootsWithMultiplicities{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmNbOfRationalRootsWithMultiplicities{<sturmname>}``

    Expands to the number of rational roots (counted with multiplicities).

    .. attention::

       :ref:`\\PolSturmIsolateZeros**{\<sturmname\>} <PolSturmIsolateZeros**>` must have been executed
       beforehand.

.. _PolSturmRationalRoot:

``\PolSturmRationalRoot{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmRationalRoot{<sturmname>}{<k>}``

    Expands to the k-th rational root.  They are enumerated from left to
    right starting at index value ``1``.

    .. attention::

       :ref:`\\PolSturmIsolateZeros**{\<sturmname\>} <PolSturmIsolateZeros**>` must have been executed
       beforehand.

.. _PolSturmRationalRootIndex:

``\PolSturmRationalRootIndex{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmRationalRootIndex{<sturmname>}{<k>}``

    Expands to the index of the ``k``\ th rational root as part of the
    ordered real roots (counted without multiplicities). So
    :ref:`\\PolSturmRationalRoot{\<sturmname\>}{\<k\>} <PolSturmRationalRoot>` is equivalent to
    this nested call::

      \PolSturmIsolatedZeroLeft{<sturmname>}{\PolSturmRationalRootIndex{<sturmname>}{<k>}}

    .. attention::

       :ref:`\\PolSturmIsolateZeros**{\<sturmname\>} <PolSturmIsolateZeros**>` must have been executed
       beforehand.

.. _PolSturmRationalRootMultiplicity:

``\PolSturmRationalRootMultiplicity{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolSturmRationalRootMultiplicity{<sturmname>}{<k>}``

    Expands to the multiplicity of the ``k``\ th rational root.

    .. attention::

       :ref:`\\PolSturmIsolateZeros**{\<sturmname\>} <PolSturmIsolateZeros**>` must have been executed
       beforehand.

.. _PolIntervalWidth:

``\PolIntervalWidth{}{}``
^^^^^^^^^^^^^^^^^^^^^^^^^

Syntax: ``\PolIntervalWidth{<sturmname>}{<index>}``

    The ``10^E`` width of the current ``index``\ -th root localization
    interval. Output is in xintfrac_ raw ``1/1[E]`` format (if not zero).

    .. renommer cela en \PolSturmIntervalWidth ?

Expandable macros for use within execution of ``\PolPrintIntervals``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

These macros are for usage within custom user redefinitions of
`\\PolPrintIntervalsKnownRoot`_, `\\PolPrintIntervalsUnknownRoot`_, or
in redefinitions of `\PolPrintIntervalsPrintExactZero`_ (used in the
default for the former) and of `\\PolPrintIntervalsPrintLeftEndPoint`_,
`\\PolPrintIntervalsPrintRightEndPoint`_ (used in the default for the
latter).

.. vieux commentaire

   Some macros formerly mentioned here got removed at 0.7:
   ``\PolPrintIntervalsTheEndPoint``,
   ``\PolIfEndPointIsPositive{T}{F}``,
   ``\PolIfEndPointIsNegative{T}{F}``,
   ``\PolIfEndPointIsZero{T}{F}``.

.. _PolPrintIntervalsTheVar:

``\PolPrintIntervalsTheVar``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Expands to the name (default ``Z``) used for representing the roots,
    which was passed as optional argument ``varname`` to
    :ref:`\\PolPrintIntervals[varname]{\<sturmname\>} <PolPrintIntervals>`.

.. _PolPrintIntervalsTheIndex:

``\PolPrintIntervalsTheIndex``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Expands to the index of the considered interval (indexing starting
    at 1 for the leftmost interval).

.. _PolPrintIntervalsTheSturmName:

``\PolPrintIntervalsTheSturmName``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Expands to the argument which was passed as ``<sturmname>`` to
    :ref:`\\PolPrintIntervals[varname]{\<sturmname\>} <PolPrintIntervals>`.

.. _PolPrintIntervalsTheLeftEndPoint:

``\PolPrintIntervalsTheLeftEndPoint``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    The left end point of the interval, as would be produced by
    `\\PolSturmIsolatedZeroLeft <PolSturmIsolatedZeroLeft_>`_ if it was
    used with arguments the Sturm chain name and interval index returned
    by `\\PolPrintIntervalsTheSturmName`_ and
    `\\PolPrintIntervalsTheIndex`_.

.. _PolPrintIntervalsTheRightEndPoint:

``\PolPrintIntervalsTheRightEndPoint``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    The right end point of the interval, as would be produced by
    `\\\PolSturmIsolatedZeroRight <PolSturmIsolatedZeroRight_>`_ for
    this Sturm chain name and index.

.. _PolPrintIntervalsTheMultiplicity:

``\PolPrintIntervalsTheMultiplicity``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    The multiplicity of the unique root within the interval of index
    `\\PolPrintIntervalsTheIndex`_. Makes sense only if the starred (or
    double-starred) variant of `\\PolSturmIsolateZeros
    <PolSturmIsolateZeros_>`_ was used earlier.


TeX Booleans (with names enacting their defaults)
-------------------------------------------------

.. _xintverbosefalse:

``\xintverbosefalse``
~~~~~~~~~~~~~~~~~~~~~

    This is actually an xintexpr_ configuration. Setting it to
    ``true`` triggers the writing of information to the log when new
    polynomial or scalar variables are defined.

    .. caution::

       The macro and variable meanings as written to the log are to be
       considered unstable and undocumented internal structures.

.. _polnewpolverbosefalse:

``\polnewpolverbosefalse``
~~~~~~~~~~~~~~~~~~~~~~~~~~

    When ``\poldef`` is used, both a variable and a function are
    defined.  The default ``\polnewpolverbosefalse`` setting suppresses
    the print-out to the log and terminal of the function macro meaning,
    as it only duplicates the information contained in the variable
    which is already printed out to the log and terminal.

    However :ref:`\\PolGenFloatVariant{\<polname\>} <PolGenFloatVariant>`
    does still print out the information relative to the polynomial
    function it defines for use in ``\xintfloateval{}`` as there is no
    float polynomial variable, only the

    function, and it is the only way to see its rounded coefficients
    (``\xintverbosefalse`` suppresses also that info).

    If set to ``true``, it overrides in both cases
    ``\xintverbosefalse``.  The setting only affects polynomial
    declarations.  Scalar variables such as those holding information on
    roots obey only the ``\xintverbose...`` setting.

    (new with ``0.8``)

.. _poltypesetallfalse:

``\poltypesetallfalse``
~~~~~~~~~~~~~~~~~~~~~~~

    If ``true``, `\\PolTypeset <PolTypeset_>`_ will also typeset the vanishing
    coefficients.


.. _poltoexprallfalse:

``\poltoexprallfalse``
~~~~~~~~~~~~~~~~~~~~~~

    If ``true``, :ref:`\\PolToExpr{\<pol. expr.\>} <PolToExpr>` and
    :ref:`\\PolToFloatExpr{\<pol. expr.\>} <PolToFloatExpr>` will also
    include the vanishing coefficients in their outputs.


Utilities
---------

.. _PolDecToString:

``\PolDecToString{}``
~~~~~~~~~~~~~~~~~~~~~

Syntax: ``\PolDecToString{decimal number}``

    This is a utility macro to print decimal numbers.  It is an alias
    for ``\xintDecToString``.

.. vieux commentaire

    It has been
    backported to xintfrac_ (release ``1.3`` of ``2018/03/01``) under
    the name ``\xintDecToString``, and the ``polexpr`` macro is simply
    now an alias to it.

..

    For example
    ``\PolDecToString{123.456e-8}`` will expand to ``0.00000123456``
    and ``\PolDecToString{123.450e-8}`` to ``0.00000123450`` which
    illustrates that trailing zeros are not trimmed.

    To trim trailing zeroes, one can use
    ``\PolDecToString{\xintREZ{#1}}``.

    Attention that a.t.t.o.w. if the argument is for example ``1/5``, the
    macro does not identify that this is in fact a number with a finite
    decimal expansion and it outputs ``1/5``.  See current xintfrac_
    documentation.

.. vieux commentaire

    The precise behaviour of this macro may evolve in future releases of
    xintexpr_.

..


.. _polexprsetup:

``\polexprsetup{key=val,...}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Serves to customize the package. Currently only two keys are
    recognized:

    - ``norr``: the postfix that :ref:`\\PolSturmIsolateZeros**{\<sturmname\>} <PolSturmIsolateZeros**>`
      should append to ``<sturmname>`` to declare the primitive polynomial
      obtained from original one after removal of all rational roots.
      The default value is ``_norr`` (standing for “no rational roots”).

    - ``sqfnorr``: the postfix that :ref:`\\PolSturmIsolateZeros**{\<sturmname\>} <PolSturmIsolateZeros**>`
      should append to ``<sturmname>`` to declare the primitive polynomial
      obtained from original one after removal of all rational roots and
      suppression of all multiplicities.
      The default value is ``_sqf_norr`` (standing for “square-free with
      no rational roots”).

    The package executes ``\polexprsetup{norr=_norr,
    sqfnorr=_sqf_norr}`` as default.


Technicalities
--------------

- Do not use the underscore ``_`` as the *first character* of a
  polynomial name, even if of catcode letter.  This may cause an
  infinite loop.

- The ``@`` is allowed in the names of polynomials, independently of
  whether it is of catcode letter or other.  In defining macros which
  will use ``\poldef`` to create polynomials it seems reasonable to
  adopt the convention that ``@`` as *first* character in polynomial
  names is to be reserved to temporary auxiliary polynomials.

  .. attention::

     Do not use ``@_`` at start of polynomial names.  This is reserved for
     internal usage by the package.

- Catcodes are set temporarily by `\\poldef <poldef;_>`_ macro to safe
  values prior to grab the polynomial expression up to the terminator
  ``;``, and also by `\\PolDef <PolDef_>`_ prior to grab the
  brace-enclosed polynomial expression.  This gives a layer of
  protection in case some package (for example the ``babel-french`` module)
  has made some characters active.  It will fail though if the whole
  thing is located inside some definition of a macro done at a time the
  characters are active.

- ..

  .. attention::

     Contrarily to ``\xintdefvar`` and ``\xintdeffunc`` from xintexpr_,
     ``\poldef`` uses a naive delimited macro to fetch up to the
     expression terminator ``";"``, hence it will be fooled if some
     ``;`` is used inside the expression (which is possible as it
     appears in some xintexpr_ constructs).  Work-around is to use curly
     braces around the inner semi-colons, or simpler to use
     ``\PolDef``.

- As a consequence of xintfrac_ addition and subtraction always using
  least common multiples for the denominators, user-chosen common
  denominators (currently) survive additions and multiplications. For
  example, this::

    \poldef P(x):= 1/2 + 2/2*x + 3/2*x^3 + 4/2*x^4;
    \poldef Q(x):= 1/3 + (2/3)x + (3/3)x^3 + (4/3)x^4;
    \poldef PQ(x):= P*Q;

  gives internally the polynomial::

    1/6+4/6*x^1+4/6*x^2+6/6*x^3+20/6*x^4+16/6*x^5+9/6*x^6+24/6*x^7+16/6*x^8

  where all coefficients have the same denominator 6. Notice though that
  ``\PolToExpr{PQ}`` outputs the ``6/6*x^3`` as ``x^3`` because (by
  default) it recognizes and filters out coefficients equal to one or
  minus one. One can use for example
  ``\PolToCSV{PQ}`` to see the internally stored coefficients.

  .. vieux commentaire

         prior to ``0.4.1``, ``polexpr`` used to temporarily patch
         during the parsing of polynomials the xintfrac_ macros. This
         patch was backported to xintexpr_ at release ``1.3``.

- :ref:`\\PolDiff{\<polname_1\>}{\<polname_2\>} <PolDiff>` always applies ``\xintPIrr`` to the
  resulting coefficients, which means that fractions are reduced to
  lowest terms but ignoring an already separated *power of ten* part ``[N]``
  present in the internal representation.  This is tentative and may change.

  Same remark for :ref:`\\PolAntiDiff{\<polname_1\>}{\<polname_2\>} <PolAntiDiff>`.

- Currently, the package stores all coefficients from index ``0`` to
  index equal to the polynomial degree inside a single macro, as a list.
  This data structure is obviously very inefficient for polynomials of
  high degree and few coefficients (as an example with ``\poldef
  f(x):=x^1000 + x^500;`` the subsequent definition ``\poldef g(x):=
  f(x)^2;`` will do of the order of 1,000,000 multiplications and
  additions involvings only zeroes... which does take time). This
  may change in the future.

- As is to be expected internal structures of the package are barely
  documented and unstable. Don't use them.


.. _xinttools:
.. _xintfrac:
.. _xint: https://www.ctan.org/pkg/xint

.. _xintexpr: https://www.ctan.org/pkg/xintexpr

.. _xintsession: https://www.ctan.org/pkg/xintsession

.. _Wilkinson polynomial: https://en.wikipedia.org/wiki/Wilkinson%27s_polynomial

.. _Sturm algorithm:
.. _Sturm Theorem: https://en.wikipedia.org/wiki/Sturm%27s_theorem

.. _DocUtils: http://docutils.sourceforge.net/docs/index.html


.. Local variables:
.. sentence-end-double-space: t;
.. End: